The OWL 2 Web Ontology Language, informally OWL 2, is an ontology language for the Semantic Web with formally defined meaning. OWL 2 ontologies provide classes, properties, individuals, and data values and are stored as Semantic Web documents. OWL 2 ontologies can be used along with information written in RDF, and OWL 2 ontologies themselves are primarily exchanged as RDF documents. The OWL 2 Document Overview describes the overall state of OWL 2, and should be read before other OWL 2 documents.

This document defines the RDF-compatible model-theoretic semantics of OWL 2.

May Be Superseded

This section describes the status of this document at the time of its publication. Other documents may supersede this document. A list of current W3C publications and the latest revision of this technical report can be found in the W3C technical reports index at http://www.w3.org/TR/.

XML Schema Datatypes Dependency

OWL 2 is defined to use datatypes defined in the XML Schema Definition Language (XSD). As of this writing, the latest W3C Recommendation for XSD is version 1.0, with version 1.1 progressing toward Recommendation. OWL 2 has been designed to take advantage of the new datatypes and clearer explanations available in XSD 1.1, but for now those advantages are being partially put on hold. Specifically, until XSD 1.1 becomes a W3C Recommendation, the elements of OWL 2 which are based on it should be considered optional, as detailed in Conformance, section 2.3. Upon the publication of XSD 1.1 as a W3C Recommendation, those elements cease to be optional and are to be considered required as otherwise specified.

We suggest that for now developers and users follow the XSD 1.1 Candidate Recommendation. Based on discussions between the Schema and OWL Working Groups, we do not expect any implementation changes will be necessary as XSD 1.1 advances to Recommendation.

Endorsed By W3C

This document has been reviewed by W3C Members, by software developers, and by other W3C groups and interested parties, and is endorsed by the Director as a W3C Recommendation. It is a stable document and may be used as reference material or cited from another document. W3C's role in making the Recommendation is to draw attention to the specification and to promote its widespread deployment. This enhances the functionality and interoperability of the Web.

Technically,
the OWL 2 RDF-Based Semantics
is defined as a
semantic extension
of
"D-Entailment"
(RDFS with datatype support),
as specified in the RDF Semantics
[RDF Semantics].
In other words,
the meaning given to an RDF graph by the OWL 2 RDF-Based Semantics
includes the meaning provided by the semantics of RDFS with datatypes,
and additional meaning is specified for all the language constructs of OWL 2,
such as Boolean connectives,
sub property chains
and qualified cardinality restrictions
(see the OWL 2 Structural Specification
[OWL 2 Specification]
for further information
on all the language constructs of OWL 2).
The definition of the semantics for the extra constructs
follows the design principles
as applied to the RDF Semantics.

The content of this document is not meant to be self-contained
but builds on top of the
RDF Semantics document
[RDF Semantics]
by adding those aspects
that are specific to OWL 2.
Hence,
the complete definition of the OWL 2 RDF-Based Semantics
is given by
the combination of both
the RDF Semantics document
and the document at hand.
In particular,
the terminology used in the RDF Semantics
is reused here
except for cases
where a conflict exists with the rest of the OWL 2 specification.

The remainder of this section
provides an overview
of some of the distinguishing features
of the OWL 2 RDF-Based Semantics
and outlines the document's structure and content.

In Section 2,
the syntax
over which the OWL 2 RDF-Based Semantics is defined
is the set of all
RDF graphs
[RDF Concepts].
The OWL 2 RDF-Based Semantics
provides a precise formal meaning
for every RDF graph.
The language
that is determined
by RDF graphs
being interpreted using the OWL 2 RDF-Based Semantics
is called
"OWL 2 Full".
In this document,
RDF graphs are also called
"OWL 2 Full ontologies",
or simply "ontologies",
unless there is risk of confusion.

The OWL 2 RDF-Based Semantics
interprets the
RDF
and
RDFS vocabularies
[RDF Semantics]
and the OWL 2 RDF-Based vocabulary
together with an extended set of datatypes
and their constraining facets
(see Section 3).

OWL 2 RDF-Based interpretations
(Section 4)
are defined on a universe
(see Section 1.3 of the RDF Semantics specification
[RDF Semantics]
for an overview of
the basic intuition of model-theoretic semantics).
The universe is divided into parts,
namely individuals, classes, and properties,
which are identified with their RDF counterparts
(see Figure 1).
The part of individuals equals the whole universe.
This means
that all classes and properties are also
individuals in their own right.
Further,
every name interpreted by an OWL 2 RDF-Based interpretation
denotes an individual.

The three basic parts are divided into further parts as follows.
The part of individuals subsumes the part of data values,
which comprises the denotations of all literals.
Also subsumed by the individuals is the part of ontologies.
The part of classes subsumes the part of datatypes,
which are classes
consisting entirely of data values.
Finally,
the part of properties subsumes the parts of
object properties,
data properties,
ontology properties
and annotation properties.
The part of object properties equals the whole part of properties,
and therefore all other kinds of properties are also object properties.

For annotations properties
note that annotations are not "semantic-free"
under the OWL 2 RDF-Based Semantics.
Just like every other triple or set of triples occurring in an RDF graph,
an annotation is assigned a truth value by any given OWL 2 RDF-Based interpretation.
Hence,
although annotations are meant to be "semantically weak",
i.e., their formal meaning does not significantly exceed
that originating from the RDF Semantics specification,
adding an annotation
may still change the meaning of an ontology.
A similar discussion holds for statements
that are built from ontology properties,
such as owl:imports,
which are used to define relationships between two ontologies.

Every class represents a specific set of individuals,
called the class extension of the class:
an individual a is an instance of a class C,
if a is a member of the class extension ICEXT(C).
Since a class is itself an individual under the OWL 2 RDF-Based Semantics,
classes are distinguished from their respective class extensions.
This distinction allows,
for example,
that a class may be an instance of itself
by being a member of its own class extension.
Also,
two classes may be equivalent
by sharing the same class extension,
although being different individuals,
e.g., they do not need to share the same properties.
Similarly,
every property has an associated property extension
that consists of pairs of individuals:
an individual a1
has a relationship to an individual a2
with respect to a property p
if the pair
( a1 , a2 )
is a member of the property extension IEXT(p).
Again, properties are distinguished from their property extensions.
In general,
if there are no further constraints,
an arbitrary extension may be associated with
a given class or property,
and two interpretations may associate
distinct extensions
with the same class or property.

Individuals may play different "roles".
For example,
an individual can be
both a data property and an annotation property,
since the different parts of the universe
of an OWL 2 RDF-Based interpretation
are not required to be mutually disjoint,
or an individual can be
both a class and a property
by associating
both a class extension and a property extension
with it.
In the latter case
there will be no specific relationship
between the class extension and the property extension
of such an individual
without further constraints.
For example,
the same individual
can have an empty class extension
while having a nonempty property extension.

The main part of the OWL 2 RDF-Based Semantics is Section 5,
which specifies
a formal meaning for all the OWL 2 language constructs
by means of the
OWL 2 RDF-Based semantic conditions.
These semantic conditions extend all the
semantic conditions given in the RDF Semantics
[RDF Semantics].
The OWL 2 RDF-Based semantic conditions effectively determine
which sets of RDF triples are assigned a specific meaning
and what this meaning is.
For example,
semantic conditions exist
that allow one to interpret the triple
"Cowl:disjointWithD"
to mean that the denotations of the IRIs
C and D
have disjoint class extensions.

There is usually no need to provide localizing information
(e.g., by means of "typing triples")
for the IRIs occurring in an ontology.
As for the RDF Semantics,
the OWL 2 RDF-Based semantic conditions have been designed
to ensure that the denotation of any IRI
will be in the appropriate part of the universe.
For example,
the RDF triple
"Cowl:disjointWithD"
is sufficient to deduce that
the denotations of the IRIs
C and D
are actually classes.
It is not necessary to explicitly add additional typing triples
"Crdf:type rdfs:Class"
and
"Drdf:type rdfs:Class"
to the ontology.

In the RDF Semantics,
this kind of "automatic localization"
was to some extent achieved by so called
"axiomatic triples"
[RDF Semantics],
such as
"rdf:type rdf:type rdf:Property"
or
"rdf:type rdfs:domain rdfs:Resource".
However,
there is no explicit normative collection
of additional axiomatic triples
for the OWL 2 RDF-Based Semantics;
instead,
the specific axiomatic aspects of the OWL 2 RDF-Based Semantics
are determined by a subset of the OWL 2 RDF-Based semantic conditions.
Section 6
discusses axiomatic triples in general
and provides an example set of axiomatic triples
that is compatible with the OWL 2 RDF-Based Semantics.

Section 7 compares
the OWL 2 RDF-Based Semantics
with the OWL 2 Direct Semantics
[OWL 2 Direct Semantics].
While
the OWL 2 RDF-Based Semantics is based on the
RDF Semantics specification
[RDF Semantics],
the OWL 2 Direct Semantics
is a description logic style semantics.
Several fundamental differences
exist between the two semantics,
but
there is also a strong relationship
basically stating that the OWL 2 RDF-Based Semantics
is able to reflect all logical conclusions
of the OWL 2 Direct Semantics.
This means that the OWL 2 Direct Semantics
can
in a sense
be regarded as a semantics subset of the OWL 2 RDF-Based Semantics.
The precise relationship is given by the
OWL 2 correspondence theorem.

Significant effort has been spent
in keeping the design of the OWL 2 RDF-Based Semantics
as close as possible
to that of the original specification of the
OWL 1 RDF-Compatible Semantics
[OWL 1 RDF-Compatible Semantics].
While this aim was achieved to a large degree,
the OWL 2 RDF-Based Semantics actually deviates from its predecessor in several aspects.
In most cases,
this is because of serious technical problems
that would have arisen
from a conservative
semantic extension.
One important change is that
while so called "comprehension conditions"
for the OWL 2 RDF-Based Semantics
(see Section 8)
still exist,
these are not part of the
normative set of semantic conditions anymore.
The OWL 2 RDF-Based Semantics also corrects several errors of OWL 1.
A list of differences between the two languages is given in
Section 9.

The italicized keywords MUST, MUST NOT, SHOULD, SHOULD NOT, and MAY are used to specify normative features of OWL 2 documents and tools, and are interpreted as specified in RFC 2119 [RFC 2119].

Figure 1: Parts Hierarchy of the OWL 2 RDF-Based Semantics
Each node is labeled with a class IRI
that represents a part of the universe
of an OWL 2 RDF-based interpretation.
Arrows point from parts to their super parts.

2 Ontologies

This section determines the syntax
for the OWL 2 RDF-Based Semantics,
and gives an overview on typical content of ontologies
for ontology management tasks.

The definition of an RDF triple
according to
Section 6.1 of RDF Concepts
[RDF Concepts]
is restricted to cases
where the subject of an RDF triple is
an IRI
or a
blank node
(Section 6.6 of RDF Concepts
[RDF Concepts]),
and where the predicate of an RDF triple is
an IRI.
As a consequence,
the definition does not treat cases,
where,
for example,
the subject of a triple is a literal
(Section 6.5 of RDF Concepts
[RDF Concepts]),
as in
"s" ex:p ex:o,
or where the predicate of a triple is a blank node,
as in
ex:s _:p ex:o.
In order to allow for interoperability
with other existing and future technologies and tools,
the document at hand
does not explicitly forbid the use of
generalized RDF graphs consisting of generalized RDF triples,
which are defined to allow for
IRIs, literals and blank nodes
to occur in the subject, predicate and object position.
Thus,
an RDF graph
MAY
contain generalized RDF triples,
but an implementation is not required to support generalized RDF graphs.
Note that every RDF graph consisting entirely of RDF triples according to
Section 6.1 of RDF Concepts
[RDF Concepts]
is also a generalized RDF graph.

Terminological notes:
The term
"OWL 2 Full"
refers to the language
that is determined
by the set of all RDF graphs
being interpreted using the OWL 2 RDF-Based Semantics.
Further,
in this document
the term
"OWL 2 Full ontology"
(or simply "ontology",
unless there is any risk of confusion)
will be used interchangeably
with the term "RDF graph".

2.2 Content of Ontologies (Informative)

While there do not exist any syntactic restrictions
on the set of RDF graphs
that can be interpreted by the OWL 2 RDF-Based Semantics,
in practice
an ontology will often contain certain kinds of constructs
that are aimed to support ontology management tasks.
Examples are
ontology headers
and
ontology IRIs,
as well as constructs that are about
versioning,
importing
and
annotating of ontologies,
including the concept of incompatibility between ontologies.

Implementations are not required
to support the IRI owl:onProperties,
but
MAY
support it
in order to realize
n-ary dataranges with arity ≥ 2
(see
Sections
7
and
8.4
of the OWL 2 Structural Specification
[OWL 2 Specification]
for further information).

Note:
The use of the IRI owl:DataRange has been deprecated as of OWL 2.
The IRI rdfs:DatatypeSHOULD
be used instead.

In this specification,
facets are used for defining datatype restrictions
(see Section 5.7).
For example,
to refer to the set of all strings of length 5
one can restrict
the datatype xsd:string
(Section 3.3)
by the facet xsd:length
and the value 5.

4 Interpretations

The OWL 2 RDF-Based Semantics provides
vocabulary interpretations and vocabulary entailment
(see Section 2.1 of the RDF Semantics
[RDF Semantics])
for the
RDF
and
RDFS
vocabularies
and for the
OWL 2 RDF-Based vocabulary.
This section defines
OWL 2 RDF-Based datatype maps
and
OWL 2 RDF-Based interpretations,
and specifies what
satisfaction of ontologies,
consistency and entailment
means under the OWL 2 RDF-Based Semantics.
In addition,
the so called "parts" of the universe
of an OWL 2 RDF-Based interpretation
are defined.

A datatype with facetsd
is a datatype that has the following additional components:

FS(d), the facet space of d, which is a set of pairs of the form ( F , v ), where F is an IRI called the constraining facet and v is an arbitrary data value called the constraining value;

F2V(d), the facet-to-value mapping of d, which maps each facet-value pair ( F , v ) in FS(d) to a subset of VS(d).

Note that
it is not further specified
what the nature of the denotation of a facet IRI is,
i.e. it is only known that a facet IRI denotes some individual.
Semantic extensions
MAY
impose further restrictions on the denotations of facets.
In fact,
Section 5.3
will define additional restrictions on facets.

Also note
that for a datatype d
and a facet-value pair ( F , v ) in FS(d)
the value v
is not required
to be included in the value space VS(d) of d itself.
For example,
the datatype xsd:string
(Section 3.3)
has the facet xsd:length
(Section 3.4),
which takes nonnegative integers as its constraining values
rather than strings.

In this document,
it will always be assumed from now on that
any datatype d is a datatype with facets.
If the facet space FS(d) of a datatype d
has not been explicitly defined,
or if it is not derived from another datatype's facet space
according to some well defined condition,
then FS(d) is the empty set.
Unless there is any risk of confusion,
the term "datatype"
will always refer to a datatype with facets.

Section 5.1 of the RDF Semantics specification
[RDF Semantics]
further
defines a datatype mapD to be
a set of name-datatype pairs
( u , d )
consisting of an IRI u and a datatype d,
such that no IRI appears twice in the set.
As a consequence of what has been said before,
in this document
every datatype map D will entirely consist of datatypes with facets.

The following definition specifies what an OWL 2 RDF-Based datatype map is.

Definition 4.1 (OWL 2 RDF-Based Datatype Map):
A datatype map D
is an OWL 2 RDF-Based datatype map,
if and only if
for every datatype name u listed in Section 3.3
and its respective set of constraining facets (Section 3.4)
there is
a name-datatype pair ( u, d ) in D
with the specified
lexical space LS(d),
value space VS(d),
lexical-to-value mapping L2V(d),
facet space FS(d) and
facet-to-value mapping F2V(d).

Note that Definition 4.1
does not prevent additional datatypes
to be in an OWL 2 RDF-Based datatype map.
For the special case of
an OWL 2 RDF-Based datatype map D
that exclusively contains the datatypes listed in
Section 3.3,
it is ensured that
there are datatypes available for all the facet values,
i.e.,
for every name-datatype pair ( u , d ) in D
and for every facet-value pair
( F , v )
in FS(d)
there exists a name-datatype pair ( u* , d* ) in D
such that v is in VS(d*).

IR is the universe of I,
i.e., a nonempty set
that contains at least
the denotations of literals and IRIs in V.
IP is a subset of IR,
the properties of I.
LV,
the data values of I,
is a subset of IR
that contains at least the set of plain literals
(see Section 6.5 of RDF Concepts
[RDF Concepts])
in V,
and
the value spaces of each datatype of D.
IEXT is used to associate properties with their property extension,
and is a mapping from IP to the powerset of IR × IR.
IS is a mapping from IRIs in V to their denotations in IR.
In particular,
IS(u) = d
for any name-datatype pair ( u , d ) in D.
IL is a mapping from typed literals
"s"^^u
in V
to their denotations in IR,
where IL("s"^^u) = L2V(d)(s),
provided that d is a datatype of D,
IS(u) = d, and
s is in the lexical space LS(d);
otherwise
IL("s"^^u)
is not in LV.

Convention:
Following the practice introduced in
Section 1.4 of the RDF Semantics
[RDF Semantics],
for a given interpretation I of a vocabulary V
the notation
"I(x)"
will be used
instead of "IL(x)" and "IS(x)"
for the typed literals and IRIs x in V,
respectively.

In this document,
the basic definition of a D-interpretation
is extended to take facets into account.

A D-interpretation with facetsI
is a D-interpretation for a datatype map D
consisting entirely of datatypes with facets
(Section 4.1),
where I meets the following additional semantic conditions:
for each name-datatype pair ( u , d ) in D
and each facet-value pair ( F , v ) in the facet space FS(d)

F is in the vocabulary V of I;

a name-datatype pair ( u* , d* ) exists in D, such that v is in the value space VS(d*).

In this document,
it will always be assumed from now on that
any D-interpretation I
is a D-interpretation with facets.
Unless there is any risk of confusion,
the term "D-interpretation"
will always refer to a D-interpretation with facets.

The following definition specifies what an OWL 2 RDF-Based interpretation is.

Definition 4.2 (OWL 2 RDF-Based Interpretation):
Let D be an OWL 2 RDF-Based datatype map,
and let V be a vocabulary
that includes
the RDF and RDFS vocabularies
and the OWL 2 RDF-Based vocabulary
together with all the datatype and facet names
listed in Section 3.
An OWL 2 RDF-Based interpretation,
I = ( IR , IP , IEXT , IS , IL , LV ),
of V with respect to D
is a D-interpretation of V with respect to D
that meets all the extra semantic conditions
given in Section 5.

4.3 Satisfaction, Consistency and Entailment

The following definitions specify
what it means for an RDF graph to be satisfied
by a given OWL 2 RDF-Based interpretation,
to be consistent
under the OWL 2 RDF-Based Semantics,
and to entail another RDF graph.

The notion of satisfaction under the OWL 2 RDF-Based Semantics
is based on the notion of satisfaction for
D-interpretations
and
Simple interpretations,
as defined in the RDF Semantics
[RDF Semantics].
In essence,
in order to satisfy an RDF graph,
an interpretation I has to satisfy all the triples in the graph,
i.e.,
for a triple "s p o"
it is necessary that the relationship
( I(s) , I(o) ) ∈ IEXT(I(p))
holds
(special treatment exists for blank nodes,
as detailed in
Section 1.5 of the RDF Semantics
[RDF Semantics]).
In other words,
the given graph has to be compatible with
the specific form of the IEXT mapping of I.
The distinguishing aspect of OWL 2 RDF-Based satisfaction is
that an interpretation I needs to meet
all the OWL 2 RDF-Based semantic conditions
(see Section 5),
which have a constraining effect
on the possible forms an IEXT mapping can have.

Definition 4.3 (OWL 2 RDF-Based Satisfaction):
Let G be an RDF graph,
let D be an OWL 2 RDF-Based datatype map,
let V be a vocabulary
that includes
the RDF and RDFS vocabularies
and the OWL 2 RDF-Based vocabulary
together with all the datatype and facet names listed in Section 3,
and let I be a D-interpretation of V with respect to D.
IOWL 2 RDF-Based satisfiesG with respect to V and D
if and only if
I is an OWL 2 RDF-Based interpretation of V with respect to D
that
satisfies G
as a D-interpretation of V with respect to D
according to the RDF Semantics
[RDF Semantics].

Definition 4.4 (OWL 2 RDF-Based Consistency):
Let S be a collection of RDF graphs,
and let D be an OWL 2 RDF-Based datatype map.
S is OWL 2 RDF-Based consistent with respect to D
if and only if
there is some OWL 2 RDF-Based interpretation I with respect to D
of some vocabulary V
that includes
the RDF and RDFS vocabularies
and the OWL 2 RDF-Based vocabulary
together with all the datatype and facet names listed in Section 3,
such that I OWL 2 RDF-Based satisfies all the RDF graphs in S
with respect to V and D.

Definition 4.5 (OWL 2 RDF-Based Entailment):
Let S1 and S2 be collections of RDF graphs,
and let D be an OWL 2 RDF-Based datatype map.
S1OWL 2 RDF-Based entailsS2 with respect to D
if and only if
for every OWL 2 RDF-Based interpretation I with respect to D
of any vocabulary V that includes
the RDF and RDFS vocabularies
and the OWL 2 RDF-Based vocabulary
together with all the datatype and facet names listed in Section 3
the following holds:
If I
OWL 2 RDF-Based satisfies all the RDF graphs in S1
with respect to V and D,
then I
OWL 2 RDF-Based satisfies all the RDF graphs in S2
with respect to V and D.

4.4 Parts of the Universe

Table 4.1
defines the "parts" of the universe
of a given OWL 2 RDF-Based interpretation I.

The second column tells the name of the part.
The third column gives a definition of the part
in terms of the mapping IEXT of I,
and by referring to a particular term
of the RDF, RDFS or OWL 2 RDF-Based vocabulary.

As an example,
the part of all datatypes is named "IDC",
and it is defined as the set of all individuals x
for which the relationship
"( x , I(rdfs:Datatype) )
∈
IEXT(I(rdf:type))"
holds.
According to the semantics of rdf:type,
as defined in
Section 4.1 of the RDF Semantics
[RDF Semantics],
this means that the name "IDC"
denotes the class extension
(see Section 4.5)
of I(rdfs:Datatype).

4.5 Class Extensions

The mapping ICEXT from IC to the powerset of IR,
which associates classes with their class extension,
is defined
for every c ∈ IC
as

ICEXT(c) = { x ∈ IR | ( x , c ) ∈ IEXT(I(rdf:type)) } .

5 Semantic Conditions

This section defines the semantic conditions of the OWL 2 RDF-Based Semantics.
The semantic conditions presented here
are basically only those for the specific constructs of OWL 2.
The complete set of semantic conditions for the OWL 2 RDF-Based Semantics
is the combination of the semantic conditions presented here
and the semantic conditions
for
Simple Entailment,
RDF,
RDFS
and
D-Entailment,
as specified in
the RDF Semantics specification
[RDF Semantics].

All semantic conditions in this section
are defined with respect to an interpretation I.
Section 5.1
specifies semantic conditions for the different parts of the universe
of the interpretation being considered
(compare Section 4.4).
Section 5.2
and
Section 5.3
list semantic conditions for the classes and the properties of the OWL 2 RDF-Based vocabulary.
In the rest of this section,
the OWL 2 RDF-Based semantic conditions
for the different language constructs of OWL 2
are specified.

Conventions used in this Section

iff:
Throughout this section
the term "iff" is used as a shortform for "if and only if".

Conjunctive commas:
A comma
(",")
separating two assertions in a semantic condition, as in
"c ∈ IC , p ∈ IP",
is read as a logical "and".
Further,
a comma separating two variables,
as in
"c, d ∈ IC",
is used for abbreviating two comma separated assertions,
"c ∈ IC , d ∈ IC"
in this example.

Unscoped variables:
If no explicit scope is given for a variable "x",
as in "∀ x : …" or "{ x | … }",
then "x" is unconstrained,
which means x ∈ IR,
i.e. "x" denotes an arbitrary individual in the universe.

Set cardinality:
For a set S,
an expression of the form "#S" means the number of elements in S.

Sequence expressions:
An expression of the form
"s sequence of a1 , … , an ∈ S"
means that "s" represents an RDF list of n ≥ 0
individuals a1 , … , an,
all of them being members of the set S.
Precisely,
s = I(rdf:nil) for n = 0;
and for n > 0
there exist
z1 ∈ IR , … , zn ∈ IR,
such that

Note, as mentioned in
Section 3.3.3 of the RDF Semantics
[RDF Semantics],
there are no semantic constraints that enforce "well-formed" sequence structures.
So, for example,
it is possible for a sequence head s to refer to more than one sequence.

Set names:
The following names are used as convenient abbreviations for certain sets:

ISEQ: The set of all sequences. This set equals the class extension of rdf:List, i.e., ISEQ := ICEXT(I(rdf:List)).

INNI: The set of all nonnegative integers. This set equals the value space of the datatype xsd:nonNegativeInteger, i.e., INNI := ICEXT(I(xsd:nonNegativeInteger)), but is also subsumed by the value spaces of other numerical datatypes, such as xsd:integer.

Further,
special treatment exists for OWL 2 axioms
that have a multi-triple representation in RDF,
where the different triples share a common "root node",
such as the blank node
"_:x"
in the following example:

In essence,
the semantic conditions for the encoding of these language constructs
are "iff" conditions,
as usual for axioms.
However,
in order to cope with the specific syntactic aspect of a "root node",
the "iff" conditions of these language constructs have been split into two "if-then" conditions,
where the "if-then" condition representing the right-to-left direction
contains an additional premise
having the form
"∃ z ∈ IR".
The purpose of this premise is to ensure the existence of an individual
that is needed to satisfy the root node
under the OWL 2 RDF-Based semantics.
The language constructs in question are
n-ary disjointness axioms
in Section 5.10,
and
negative property assertions
in Section 5.15.

The "if-then" semantic conditions in this section
sometimes do not explicitly list all typing statements in their consequent
that one might expect.
For example,
the semantic condition for
owl:someValuesFrom restrictions in
Section 5.6
does not list the statement
"x ∈ ICEXT(I(owl:Restriction))"
on its right hand side.
Consequences are generally not mentioned,
if they can already be deduced by other means.
Often,
these redundant consequences follow from the
semantic conditions for
vocabulary classes and vocabulary properties
in
Section 5.2
and
Section 5.3,
respectively,
occasionally in connection with the semantic conditions
for the parts of the universe
in
Section 5.1.
In the example above,
the omitted consequence can be obtained
from the third column of the entry for
owl:someValuesFrom
in the table in
Section 5.3,
which determines that
IEXT(I(owl:someValuesFrom))
⊆
ICEXT(I(owl:Restriction)) × IC.

5.1 Semantic Conditions for the Parts of the Universe

Table 5.1
lists the semantic conditions
for the parts of the universe
of the OWL 2 RDF-Based interpretation being considered.
Additional semantic conditions affecting these parts
are given in Section 5.2.

The first column tells the name of the part,
as defined in
Section 4.4.
The second column defines
certain conditions on the part.
In most cases,
the column specifies for the part
by which other part it is subsumed,
and thus the position of the part
in the "parts hierarchy" of the universe
is narrowed down.
The third column provides further
information about the instances
of those parts
that consist of classes or properties.
In general,
if the part consists of classes,
then for the class extensions of the member classes
is specified by which part of the universe they are subsumed.
If the part consists of properties,
then the domains and ranges of the member properties are determined.

Table 5.1: Semantic Conditions for the Parts of the Universe

Name ofPart S

Conditions on S

Conditions onInstances x of S

IR

S ≠ ∅

LV

S ⊆ IR

IX

S ⊆ IR

IC

S ⊆ IR

ICEXT(x) ⊆ IR

IDC

S ⊆ IC

ICEXT(x) ⊆ LV

IP

S ⊆ IR

IEXT(x) ⊆ IR × IR

IODP

S ⊆ IP

IEXT(x) ⊆ IR × LV

IOXP

S ⊆ IP

IEXT(x) ⊆ IX × IX

IOAP

S ⊆ IP

IEXT(x) ⊆ IR × IR

5.2 Semantic Conditions for the Vocabulary Classes

Table 5.2
lists the semantic conditions for the classes
that have IRIs in the OWL 2 RDF-Based vocabulary.
In addition,
the table contains all those classes
with IRIs in the RDF and RDFS vocabularies
that represent
parts of the universe
of the OWL 2 RDF-Based interpretation being considered
(Section 4.4).
The semantic conditions for the remaining classes
with names in the
RDF
and
RDFS vocabularies
can be found in the RDF Semantics specification
[RDF Semantics].

The first column tells the IRI of the class.
The second column defines
of what particular kind a class is,
i.e. whether it is a general class (a member of the part IC)
or a datatype (a member of IDC).
The third column specifies
for the class extension of the class
by which part of the universe
(Section 4.4)
it is subsumed:
from an entry of the form
"ICEXT(I(C)) ⊆ S",
for a class IRI C
and a set S,
and given an RDF triple of the form
"urdf:typeC",
one can deduce
that the relationship
"I(u) ∈ S"
holds.
Note that some entries are of the form
"ICEXT(I(C)) = S",
which means that the class extension is exactly specified to be that set.
See Section 5.1
for further semantic conditions
on those classes that represent parts.

Not included in this table are the datatypes of the OWL 2 RDF-Based Semantics
with IRIs listed in Section 3.3.
For each such datatype IRI E,
the following semantic conditions hold
(as a consequence of
the fact that E is a member of the datatype map
of every OWL 2 RDF-Based interpretation
according to
Definition 4.2,
and by the "general semantic conditions for datatypes"
listed in
Section 5.1 of the RDF Semantics
[RDF Semantics]):

I(E) ∈ IDC

ICEXT(I(E)) ⊆ LV

Table 5.2: Semantic Conditions for the Vocabulary Classes

IRI E

I(E)

ICEXT(I(E))

owl:AllDifferent

∈ IC

⊆ IR

owl:AllDisjointClasses

∈ IC

⊆ IR

owl:AllDisjointProperties

∈ IC

⊆ IR

owl:Annotation

∈ IC

⊆ IR

owl:AnnotationProperty

∈ IC

= IOAP

owl:AsymmetricProperty

∈ IC

⊆ IP

owl:Axiom

∈ IC

⊆ IR

rdfs:Class

∈ IC

= IC

owl:Class

∈ IC

= IC

owl:DataRange

∈ IC

= IDC

rdfs:Datatype

∈ IC

= IDC

owl:DatatypeProperty

∈ IC

= IODP

owl:DeprecatedClass

∈ IC

⊆ IC

owl:DeprecatedProperty

∈ IC

⊆ IP

owl:FunctionalProperty

∈ IC

⊆ IP

owl:InverseFunctionalProperty

∈ IC

⊆ IP

owl:IrreflexiveProperty

∈ IC

⊆ IP

rdfs:Literal

∈ IDC

= LV

owl:NamedIndividual

∈ IC

⊆ IR

owl:NegativePropertyAssertion

∈ IC

⊆ IR

owl:Nothing

∈ IC

= ∅

owl:ObjectProperty

∈ IC

= IP

owl:Ontology

∈ IC

= IX

owl:OntologyProperty

∈ IC

= IOXP

rdf:Property

∈ IC

= IP

owl:ReflexiveProperty

∈ IC

⊆ IP

rdfs:Resource

∈ IC

= IR

owl:Restriction

∈ IC

⊆ IC

owl:SymmetricProperty

∈ IC

⊆ IP

owl:Thing

∈ IC

= IR

owl:TransitiveProperty

∈ IC

⊆ IP

5.3 Semantic Conditions for the Vocabulary Properties

Table 5.3
lists the semantic conditions for the properties
that have IRIs in the OWL 2 RDF-Based vocabulary.
In addition,
the table contains all those properties
with IRIs in the RDFS vocabulary
that are specified to be annotation properties
under the OWL 2 RDF-Based Semantics.
The semantic conditions for the remaining properties
with names in the
RDF
and
RDFS
vocabularies
can be found in the RDF Semantics specification
[RDF Semantics].

The first column tells the IRI of the property.
The second column defines
of what particular kind a property is,
i.e. whether it is a general property (a member of the part IP),
a datatype property (a member of IODP),
an ontology property (a member of IOXP) or
an annotation property (a member of IOAP).
The third column specifies
the domain and range of the property:
from an entry of the form
"IEXT(I(p)) ⊆ S1 × S2",
for a property IRI p
and sets S1 and S2,
and given an RDF triple
"spo",
one can deduce the relationships
"I(s) ∈ S1"
and
"I(o) ∈ S2".
Note that some entries are of the form
"IEXT(I(p)) = S1 × S2",
which means that the property extension is exactly specified
to be the Cartesian product of the two sets.

Not included in this table are the facets of the OWL 2 RDF-Based Semantics
with IRIs
listed in Section 3.4,
which are used to specify datatype restrictions
(see Section 5.7).
For each such facet IRI E,
the following semantic conditions
extend
the basic semantics specification
that has been given for
datatypes with facets
in Section 4.1:

I(E) ∈ IODP

IEXT(I(E)) ⊆ IR × LV

Implementations are not required
to support the semantic condition for
owl:onProperties,
but
MAY
support it
in order to realize
n-ary dataranges with arity ≥ 2
(see
Sections
7
and
8.4
of the OWL 2 Structural Specification
[OWL 2 Specification]
for further information).

Informative notes:

owl:topObjectProperty
relates every two individuals in the universe with each other.
Likewise, owl:topDataProperty
relates every individual with every data value.
Further,
owl:bottomObjectProperty
and
owl:bottomDataProperty
stand both for the empty relationship.

The ranges of the properties
owl:deprecated and owl:hasSelf
are not restricted in any form,
and, in particular,
they are not restricted to Boolean values.
The actual object values of these properties
do not have any intended meaning,
but could as well have been defined to be of any other value.
Therefore, the semantics given here are of a form
that the values can be arbitrarily chosen
without leading to any nontrivial semantic conclusions.
It is, however, recommended to still use an object literal of the form
"true"^^xsd:boolean
in ontologies,
in order to not get in conflict
with the required usage of these properties
in scenarios that ask for applying the reverse RDF mapping
(compare Table 13 in
Section 3.2.4 of the OWL 2 RDF Mapping
[OWL 2 RDF Mapping]
for owl:hasSelf,
and
Section 5.5 of the OWL 2 Structural Specification
[OWL 2 Specification]
for owl:deprecated).

The range of the property
owl:annotatedProperty
is unrestricted,
i.e. it is not specified as the set of properties.
Annotations are meant to be "semantically weak",
i.e. their formal meaning should not significantly exceed
that originating from the RDF Semantics specification.

5.4 Semantic Conditions for Boolean Connectives

Table 5.4
lists the semantic conditions for Boolean connectives,
including
intersections, unions and complements
of classes and datatypes.
An intersection or a union of a collection of datatypes
or a complement of a datatype
is itself a datatype.
While a complement of a class is created w.r.t. the whole universe,
a datatype complement is created for a datatype w.r.t. the set of data values only.

Informative notes:
Of the three pairs of semantic conditions in the table
every first is an "iff" condition,
since the corresponding OWL 2 language constructs
are both
class expressions and axioms.
In contrast,
the semantic condition on datatype complements
is an "if-then" condition,
since it only corresponds to a datarange expression.
See the
notes on the form of semantic conditions
for further information.
For the remaining semantic conditions
that treat the cases of intersections and unions of datatypes
it is sufficient to have "if-then" conditions,
since stronger "iff" conditions would be redundant
due to the more general "iff" conditions
that already exist for classes.
Note that the datatype related semantic conditions
do not apply to empty sets,
but one can still receive a datatype from an empty set
by explicitly asserting the resulting class
to be an instance of class rdfs:Datatype.

5.5 Semantic Conditions for Enumerations

Table 5.5
lists the semantic conditions for enumerations,
i.e. classes that consist of an explicitly given finite set of instances.
In particular, an enumeration entirely consisting of data values is a datatype.

Informative notes:
The first semantic condition is an "iff" condition,
since the corresponding OWL 2 language construct
is both a class expression and an axiom.
See the
notes on the form of semantic conditions
for further information.
For the remaining semantic condition
that treats the case of enumerations of data values
it is sufficient to have an "if-then" condition,
since a stronger "iff" condition would be redundant
due to the more general "iff" condition
that already exists for individuals.
Note that the data value related semantic condition
does not apply to empty sets,
but one can still receive a datatype from an empty set
by explicitly asserting the resulting class
to be an instance of class rdfs:Datatype.

5.6 Semantic Conditions for Property Restrictions

Value restrictions require that
some or all of the values of a certain property
must be instances of a given class or data range,
or that the property has a specifically defined value.
By placing a self restriction on some given property
one only considers those individuals
that are reflexively related to themselves via this property.
Cardinality restrictions determine
how often a certain property is allowed
to be applied to a given individual.
Qualified cardinality restrictions
are more specific than cardinality restrictions
in that they determine the quantity of a property application
with respect to a particular class or data range
from which the property values are taken.

Implementations are not required
to support the semantic conditions for
owl:onProperties,
but
MAY
support them
in order to realize
n-ary dataranges with arity ≥ 2
(see
Sections
7
and
8.4
of the OWL 2 Structural Specification
[OWL 2 Specification]
for further information).

Informative notes:
All the semantic conditions are "if-then" conditions,
since the corresponding OWL 2 language constructs
are class expressions.
The "if-then" conditions generally only list those consequences
on their right hand side
that are specific for the respective condition,
i.e. consequences that do not already follow by other means.
See the
notes on the form of semantic conditions
for further information.
Note that the semantic condition for self restrictions
does not constrain the right hand side of
a owl:hasSelf assertion
to be the Boolean value "true"^^xsd:boolean.
See Section 5.3 for an explanation.

5.7 Semantic Conditions for Datatype Restrictions

Table 5.7
lists the semantic conditions for datatype restrictions,
which are used to define sub datatypes of existing datatypes
by restricting the original datatype
by means of a set of facet-value pairs.
See Section 3.4
for information and an example on constraining facets.

Certain special cases exist:
If no facet-value pair is applied to a given datatype,
then the resulting datatype will be equivalent to the original datatype.
Further,
if a facet-value pair is applied to a datatype
without being a member of the datatype's facet space,
then the ontology cannot be satisfied
and will therefore be inconsistent.
In particular,
a datatype restriction with one or more specified facet-value pairs
will result in an inconsistent ontology,
if applied to a datatype with an empty facet space.

The set IFS
is defined by
IFS(d) := { ( I(F) , v ) | ( F , v ) ∈ FS(d) } ,
where
d is a datatype,
F is the IRI of a constraining facet,
and v is a constraining value of the facet.
This set corresponds to the facet space FS(d),
as defined in Section 4.1,
but rather consists of
pairs of the denotation of a facet and a value.

The mapping IF2V
is defined by
IF2V(d)(( I(F) , v )) := F2V(d)(( F , v )) ,
where
d is a datatype,
F is the IRI of a constraining facet,
and v is a constraining value of the facet.
This mapping corresponds to the facet-to-value mapping F2V(d),
as defined in Section 4.1,
resulting in the same subsets of the value space VS(d),
but rather applies to
pairs of the denotation of a facet and a value.

Informative notes:
The semantic condition is an "if-then" condition,
since the corresponding OWL 2 language construct
is a datarange expression.
The "if-then" condition only lists those consequences
on its right hand side
that are specific for the condition,
i.e. consequences that do not already follow by other means.
See the
notes on the form of semantic conditions
for further information.

5.8 Semantic Conditions for the RDFS Vocabulary

Table 5.8extends the RDFS semantic conditions
for subclass axioms, subproperty axioms, domain axioms and range axioms.
The semantic conditions provided here are "iff" conditions,
while the original semantic conditions,
as specified in
Section 4.1 of the RDF Semantics
[RDF Semantics],
are weaker "if-then" conditions.
Only the additional semantic conditions are given here
and the other conditions of RDF and RDFS
are retained.

Informative notes:
All the semantic conditions are "iff" conditions,
since the corresponding OWL 2 language constructs
are axioms.
See the
notes on the form of semantic conditions
for further information.

Table 5.8: Semantic Conditions for the RDFS Vocabulary

( c1 , c2 ) ∈ IEXT(I(rdfs:subClassOf))

iff

c1 , c2 ∈ IC ,ICEXT(c1) ⊆ ICEXT(c2)

( p1 , p2 ) ∈ IEXT(I(rdfs:subPropertyOf))

p1 , p2 ∈ IP ,IEXT(p1) ⊆ IEXT(p2)

( p , c ) ∈ IEXT(I(rdfs:domain))

p ∈ IP , c ∈ IC ,∀ x , y : ( x , y ) ∈ IEXT(p) implies x ∈ ICEXT(c)

( p , c ) ∈ IEXT(I(rdfs:range))

p ∈ IP , c ∈ IC ,∀ x , y : ( x , y ) ∈ IEXT(p) implies y ∈ ICEXT(c)

5.9 Semantic Conditions for Equivalence and Disjointness

Table 5.9
lists the semantic conditions for specifying
that two individuals are equal or different from each other,
and that either two classes or two properties
are equivalent or disjoint with each other,
respectively.
The
property owl:equivalentClass
is also used to formulate datatype definitions
(see Section 9.4 of the OWL 2 Structural Specification
[OWL 2 Specification]
for information about datatype definitions).
In addition,
the table treats disjoint union axioms.

Informative notes:
All the semantic conditions are "iff" conditions,
since the corresponding OWL 2 language constructs
are axioms.
See the
notes on the form of semantic conditions
for further information.

5.10 Semantic Conditions for N-ary Disjointness

Table 5.10
lists the semantic conditions for specifying
n-ary diversity and disjointness axioms,
i.e. that several given individuals
are mutually different from each other,
and that several given classes or properties
are mutually disjoint with each other,
respectively.

Note that there are two alternative ways
to specify owl:AllDifferent axioms,
by using either the property
owl:members
that is used for all other constructs, too,
or by applying the legacy property
owl:distinctMembers.
Both variants have an equivalent formal meaning.

Informative notes:
The semantic conditions essentially represent "iff" conditions,
since the corresponding OWL 2 language constructs
are axioms.
However,
there are actually two semantic conditions for each language construct
due to the multi-triple RDF encoding of these language constructs.
The "if-then" conditions only list those consequences
on their right hand side
that are specific for the respective condition,
i.e. consequences that do not already follow by other means.
See the
notes on the form of semantic conditions
for further information.

5.11 Semantic Conditions for Sub Property Chains

As an example,
one can define a sub property chain axiom
that specifies
the chain consisting of the property extensions
of properties
ex:hasFather
and
ex:hasBrother
to be a sub relation of
the extension of the property
ex:hasUncle.

Informative notes:
The semantic condition is an "iff" condition,
since the corresponding OWL 2 language construct
is an axiom.
See the
notes on the form of semantic conditions
for further information.
The semantics has been specified in a way
such that a sub property chain axiom can be satisfied
without requiring the existence of a property
that has the property chain as its property extension.

5.12 Semantic Conditions for Inverse Properties

Table 5.12
lists the semantic conditions for inverse property axioms.
The inverse of a given property
is the corresponding property
with subject and object swapped
for each property assertion built from it.

Informative notes:
The semantic condition is an "iff" condition,
since the corresponding OWL 2 language construct
is an axiom.
See the
notes on the form of semantic conditions
for further information.

Table 5.12: Semantic Conditions for Inverse Properties

( p1 , p2 ) ∈ IEXT(I(owl:inverseOf))

iff

p1 , p2 ∈ IP ,IEXT(p1) = { ( x , y ) | ( y , x ) ∈ IEXT(p2) }

5.13 Semantic Conditions for Property Characteristics

If a property is functional,
then at most one distinct value can be assigned
to any given individual
via this property.
An inverse functional property can be regarded as a "key" property,
i.e. no two different individuals
can be assigned the same value
via this property.
A reflexive property relates every individual in the universe to itself,
whereas an irreflexive property does not relate any individual with itself.
If two individuals are related by a symmetric property,
then this property also relates them reversely,
while this is never the case for an asymmetric property.
A transitive property
that relates an individual a with an individual b,
and the latter with an individual c,
also relates a with c.

Informative notes:
All the semantic conditions are "iff" conditions,
since the corresponding OWL 2 language constructs
are axioms.
See the
notes on the form of semantic conditions
for further information.

5.14 Semantic Conditions for Keys

Keys provide an alternative to inverse functional properties
(see Section 5.13).
They allow for defining a property as a key local to a given class:
the specified property
will have the features of a key
only for individuals being instances of the class,
and no assumption is made
about individuals
for which membership of the class cannot be entailed.
Further,
it is possible to define "compound keys",
i.e. several properties can be combined into a single key
applicable to composite values.
Note that
keys are not functional by default
under the OWL 2 RDF-Based Semantics.

Informative notes:
The semantic condition is an "iff" condition,
since the corresponding OWL 2 language construct
is an axiom.
See the
notes on the form of semantic conditions
for further information.

5.15 Semantic Conditions for Negative Property Assertions

Table 5.15
lists the semantic conditions for negative property assertions.
They allow to state that
two given individuals are not related by a given property.

The second form based on owl:targetValue
is more specific than the first form based on owl:targetIndividual
in that the second form is restricted
to the case of negative data property assertions.
Note that the second form
will coerce the target value of a negative property assertion
into a data value,
due to the range defined for the property
owl:targetValue
in
Section 5.3.

Informative notes:
The semantic conditions essentially represent "iff" conditions,
since the corresponding OWL 2 language constructs
are axioms.
However,
there are actually two semantic conditions for each language construct,
due to the multi-triple RDF encoding of these language constructs.
The "if-then" conditions only list those consequences
on their right hand side
that are specific for the respective condition,
i.e. consequences that do not already follow by other means.
See the
notes on the form of semantic conditions
for further information.

6 Appendix: Axiomatic Triples (Informative)

The RDF Semantics specification
[RDF Semantics]
defines so called "axiomatic triples"
as part of the semantics of
RDF
and
RDFS.
Unlike the RDF Semantics,
the OWL 2 RDF-Based Semantics does not normatively specify any axiomatic triples,
since one cannot expect to find a set of RDF triples
that fully captures all "axiomatic aspects"
of the OWL 2 RDF-Based Semantics.
Furthermore,
axiomatic triples for the OWL 2 RDF-Based Semantics could,
in principle,
contain arbitrarily complex class expressions,
e.g. the union of several classes,
and by this it becomes nonobvious
which of several possible nonequivalent sets of axiomatic triples
should be selected.
However,
the OWL 2 RDF-Based Semantics includes many semantic conditions
that can in a sense be regarded as being "axiomatic",
and thus can be considered a replacement for the missing axiomatic triples.
After an overview on axiomatic triples for RDF and RDFS
in Section 6.1,
Sections 6.2
and
6.3
will discuss how the "axiomatic" semantic conditions
of the OWL 2 RDF-Based Semantics
relate to axiomatic triples.
Based on this discussion,
an explicit example set of axiomatic triples
that is compatible with the OWL 2 RDF-Based Semantics
will be provided in
Section 6.4.

6.1 Axiomatic Triples in RDF

In
RDF
and
RDFS
[RDF Semantics],
axiomatic triples are used
to provide basic meaning
for all the vocabulary terms
of the two languages.
This formal meaning is independent of any given RDF graph,
and it even holds for vocabulary terms,
which do not occur in a graph
that is interpreted by an RDF or RDFS interpretation.
As a consequence,
all the axiomatic triples of RDF and RDFS
are entailed by the empty graph,
when being interpreted under the semantics of RDF or RDFS,
respectively.

As shown by these examples,
axiomatic triples are typically used by the RDF Semantics specification
to determine the part of the universe
to which the denotation of a vocabulary term belongs (1).
In the case of a property,
the domain (2) and range (3) is specified as well.
Also, in some cases,
hierarchical relationships
between classes (4) or properties (5) of the vocabulary
are determined.

Under the OWL 2 RDF-Based Semantics,
all the axiomatic triples of RDF and RDFS
could, in principle, be replaced by
"axiomatic" semantic conditions
that have neither premises nor bound variables.
By applying the RDFS semantic conditions
given in Section 5.8,
the example axiomatic triples (1) – (5)
can be equivalently restated as:

All the axiomatic triples of RDF and RDFS
can be considered "simple"
in the sense that
they have in their object position
only single terms
from the RDF and RDFS vocabularies,
and no complex class or property expressions
appear there.

6.2 Axiomatic Triples for the Vocabulary Classes

The semantic conditions for vocabulary classes
in Section 5.2
can be considered as corresponding to
a set of axiomatic triples
for the classes in the vocabulary of the OWL 2 RDF-Based Semantics.

First,
for each IRI E
occurring in the first column of Table 5.2,
if the second column contains an entry
of the form
"I(E) ∈ S"
for some set S,
then this entry corresponds to an RDF triple of the form
"Erdf:typeC",
where C is the IRI of a vocabulary class with ICEXT(I(C)) = S.
In the table, S will always be either
the part IC of all classes,
or some sub part of IC.
Hence, in a corresponding RDF triple the IRI C will be
one of
"rdfs:Class",
"owl:Class"
(S=IC in both cases)
or "rdfs:Datatype" (S=IDC).

For example,
for the IRI "owl:FunctionalProperty",
the semantic condition

I(owl:FunctionalProperty) ∈ IC

has the corresponding axiomatic triple

owl:FunctionalProperty rdf:type rdfs:Class .

Further,
for each IRI E in the first column of the table,
if the third column contains an entry
of the form
"ICEXT(I(E)) ⊆ S"
(or "ICEXT(I(E)) = S")
for some set S,
then this entry corresponds to an RDF triple of the form
"Erdfs:subClassOfC"
(or additionally "Crdfs:subClassOfE"),
where C is the IRI of a vocabulary class with ICEXT(I(C)) = S.
In each case,
S will be one of
the parts of the universe of I.

For example,
the semantic condition

ICEXT(I(owl:FunctionalProperty)) ⊆ IP

has the corresponding axiomatic triple

owl:FunctionalProperty rdfs:subClassOf rdf:Property .

In addition,
the semantic conditions for the
parts of the universe
in Table 5.1
of Section 5.1
have to be taken into account.
In particular,
if an entry in the second column of Table 5.1
is of the form
"S1 ⊆ S2"
for some sets S1 and S2,
then this corresponds to an RDF triple
of the form
"C1owl:subClassOfC2",
where
C1 and C2
are the IRIs of vocabulary classes with
ICEXT(I(C1)) = S1
and
ICEXT(I(C2)) = S2,
respectively,
according to
Section 5.2.

Section 5.2
also specifies semantic conditions
for all the datatypes of the OWL 2 RDF-Based Semantics,
as listed in Section 3.3.
For each datatype IRI E,
such as E := "xsd:string",
for the semantic conditions
"I(E) ∈ IDC"
and
"ICEXT(I(E)) ⊆ LV"
the corresponding axiomatic triples are of the form

Erdf:type rdfs:Datatype .Erdfs:subClassOf rdfs:Literal .

In analogy to
Section 6.1
for the RDF axiomatic triples,
all the axiomatic triples for the vocabulary classes
(including datatypes)
can be considered "simple"
in the sense that
they will have in their object position
only single terms
from the RDF, RDFS and OWL 2 RDF-Based vocabularies
(Section 3.2).

Note that some of the axiomatic triples obtained in this way
already follow from the semantics of
RDF
and
RDFS,
as defined in
the RDF Semantics [RDF Semantics].

6.3 Axiomatic Triples for the Vocabulary Properties

The semantic conditions for vocabulary properties
in Section 5.3
can be considered as corresponding to
a set of axiomatic triples
for the properties in the vocabulary of the OWL 2 RDF-Based Semantics.

First,
for each IRI E
occurring in the first column of Table 5.3,
if the second column contains an entry
of the form
"I(E) ∈ S" for some set S,
then this entry corresponds to an RDF triple of the form
"Erdf:typeC",
where C is the IRI of a vocabulary class with ICEXT(I(C)) = S.
In the table,
S will always be either
the part IP of all properties,
or some sub part of IP.
Hence, in a corresponding RDF triple the IRI C will be
one of
"rdf:Property",
"owl:ObjectProperty",
(S=IP in both cases),
"owl:DatatypeProperty" (S=IODP),
"owl:OntologyProperty" (S=IOXP)
or "owl:AnnotationProperty" (S=IOAP).

For example,
for the IRI "owl:disjointWith",
the semantic condition

I(owl:disjointWith) ∈ IP

has the corresponding axiomatic triple

owl:disjointWith rdf:type rdf:Property .

Further,
for each IRI E in the first column of the table,
if the third column contains an entry
of the form
"IEXT(I(E)) ⊆ S1 × S2"
for some sets S1 and S2,
then this entry corresponds to RDF triples of the form
"Erdfs:domainC1"
and
"Erdfs:rangeC2",
where C1 and C2
are the IRIs of vocabulary classes with
ICEXT(I(C1)) = S1
and
ICEXT(I(C2)) = S2,
respectively.
Note that the sets S1 and S2
do not always correspond
to any of the parts of the universe of I.

Exceptions are the semantic conditions
"IEXT(I(owl:topObjectProperty)) = IR × IR"
and
"IEXT(I(owl:topDataProperty)) = IR × LV",
since the exactly specified property extensions of these properties
cannot be expressed solely by domain and range axiomatic triples.
For example,
the domain and range axiomatic triples for
owl:sameAs
are equal to those for
owl:topObjectProperty,
but the property extension of
owl:sameAs
is different from the property extension of
owl:topObjectProperty.

Section 5.3
also specifies semantic conditions
for all the facets of the OWL 2 RDF-Based Semantics,
as listed in Section 3.4.
For each facet IRI E,
such as E := "xsd:length",
for the semantic conditions
"I(E) ∈ IODP"
and
"IEXT(I(E)) ⊆ IR × LV"
the corresponding axiomatic triples are of the form

In analogy to
Section 6.1
for the RDF axiomatic triples,
all the axiomatic triples for the vocabulary properties
(including facets)
can be considered "simple"
in the sense that
they will have in their object position
only single terms
from the RDF, RDFS and OWL 2 RDF-Based vocabularies
(Section 3.2).

6.4 A Set of Axiomatic Triples

This section provides a concrete example set of axiomatic triples
based on the discussion in the Sections
6.2
and
6.3.
The axiomatic triples are grouped by different tables
for the classes
and the properties
of the OWL 2 RDF-Based vocabulary,
for the datatypes
and the facets
of the OWL 2 RDF-Based Semantics,
and for some of the
classes and properties of the RDFS vocabulary.
Note that this set of axiomatic triples
is not meant to be free of redundancy.

Table 6.1: Axiomatic Triples for the Classes of the OWL 2 RDF-Based Vocabulary

7 Appendix: Relationship to the Direct Semantics (Informative)

This section compares
the OWL 2 RDF-Based Semantics
with the
OWL 2 Direct Semantics
[OWL 2 Direct Semantics].
While
the OWL 2 RDF-Based Semantics is based on the
RDF Semantics specification
[RDF Semantics],
the OWL 2 Direct Semantics
is a description logic style semantics.
Several fundamental differences
exist between the two semantics,
but
there is also a strong relationship
basically stating that the OWL 2 RDF-Based Semantics is able
to reflect all logical conclusions
of the OWL 2 Direct Semantics.
This means that the OWL 2 Direct Semantics
can
in a sense
be regarded as a semantics subset of the OWL 2 RDF-Based Semantics.

Technically,
the comparison will be performed
by comparing the sets of entailments
that hold for each of the two semantics,
respectively.
The definition of an OWL 2 RDF-Based entailment
was given in
Section 4.3
of this document,
while the definition of an OWL 2 Direct entailment
is provided in
Section 2.5 of the OWL 2 Direct Semantics
[OWL 2 Direct Semantics].
In both cases,
entailments are defined for pairs of ontologies,
and such an ordered pair of two ontologies will be called an
entailment query
in this section.

Comparing the two semantics by means of entailments
will only be meaningful
if the entailment queries
allow for applying
both
the OWL 2 RDF-Based Semantics
and the
OWL 2 Direct Semantics
to them.
In order to ensure this,
the comparison will be restricted to entailment queries,
for which the left-hand side and right-hand side ontologies
are both
OWL 2 DL ontologies in RDF graph form.
These are RDF graphs that,
by applying the
reverse RDF mapping
[OWL 2 RDF Mapping],
can be transformed
into corresponding
OWL 2 DL ontologies in Functional Syntax form
according to the functional style syntax defined in the
OWL 2 Structural Specification
[OWL 2 Specification],
and which must further meet
all the restrictions on OWL 2 DL ontologies
that are specified in
Section 3 of the OWL 2 Structural Specification
[OWL 2 Specification].
In fact,
these restrictions must be mutually met by
both ontologies that occur in an entailment query,
i.e.
all these restrictions need to be satisfied
as if the two ontologies would be part of a single ontology.
Any entailment query that adheres to the conditions defined here
will be called an
OWL 2 DL entailment query.

Ideally,
the relationship between
the OWL 2 RDF-Based Semantics and the OWL 2 Direct Semantics
would be of the form that
every OWL 2 DL entailment query
that is an OWL 2 Direct entailment
is also an OWL 2 RDF-Based entailment.
However,
this desirable relationship
cannot hold in general
due to a variety of differences
that exist between
the OWL 2 RDF-Based Semantics
and the OWL 2 Direct Semantics,
as demonstrated in
Section 7.1.

Fortunately,
the problems resulting from these semantic differences
can be overcome
in a way that
for every OWL 2 DL entailment query
there is another one
for which
the desired entailment relationship indeed holds,
and the new entailment query is
semantically equivalent to the original entailment query
under the OWL 2 Direct Semantics.
This is the gist of the
OWL 2 correspondence theorem,
which will be presented in
Section 7.2.
The
proof of this theorem,
given in Section 7.3,
will further demonstrate
that such a substitute OWL 2 DL entailment query
can always be algorithmically constructed
by means of simple syntactic transformations.

7.1 Example on Semantic Differences

This section will show
that differences exist
between
the OWL 2 RDF-Based Semantics and the OWL 2 Direct Semantics,
and it will be demonstrated
how these semantic differences
complicate a comparison
of the two semantics
in terms of entailments.
An example OWL 2 DL entailment query will be given,
which will happen to be an OWL 2 Direct entailment
without being an OWL 2 RDF-Based entailment.
The section will explain
the different reasons
and will provide a resolution
of each of them.
It will turn out
that the example entailment query
can be syntactically transformed
into another
OWL 2 DL entailment query
that is both
an OWL 2 Direct entailment
and
an OWL 2 RDF-Based entailment,
while being semantically unchanged
compared to the original entailment query
under the OWL 2 Direct Semantics.
This example will motivate
the OWL 2 correspondence theorem
in Section 7.2
and its proof
in Section 7.3.

Both G1* and G2*
are
OWL 2 DL ontologies in RDF graph form
and can therefore be mapped by the
reverse RDF mapping
[OWL 2 RDF Mapping]
to the following two OWL 2 DL ontologies in Functional Syntax form
F(G1*) and F(G2*):

Note that
F(G1*) and F(G2*)
mutually meet the restrictions on OWL 2 DL ontologies
as specified in
Section 3 of the OWL 2 Structural Specification
[OWL 2 Specification].
For example,
none of the IRIs being declared as a class in F(G1*)
is declared as a datatype in F(G2*),
since this would not be allowed for an OWL 2 DL entailment query.

It follows that
F(G1*) OWL 2 Direct entails F(G2*).
To show this,
only the axioms
(4) of F(G1*)
and
(5) of F(G2*)
have to be considered.
None of the other statements in the two ontologies
are relevant for this OWL 2 Direct entailment to hold,
since they do not have a formal meaning
under the OWL 2 Direct Semantics.
However,
it turns out that the RDF graph
G1*
does not OWL 2 RDF-Based entail
G2*,
for reasons discussed in detail now.

Reason 1: An Annotation in F(G2*).
The ontology F(G2*)
contains an annotation (6).
The OWL 2 Direct Semantics
does not give a formal meaning to
annotations.
In contrast,
under the OWL 2 RDF-Based Semantics
every RDF triple occurring in an RDF graph
has a formal meaning,
including
the corresponding annotation triple (8) in G2*.
Since this annotation triple
only occurs in G2*
but not in G1*,
there will exist OWL 2 RDF-Based interpretations
that satisfy G1*
without satisfying triple (8) of G2*.
Hence,
G1*
does not OWL 2 RDF-Based entail G2*.

Resolution of Reason 1.
The annotation triple (8) in G2*
will be removed,
which will avoid requiring
OWL 2 RDF-Based interpretations to interpret this triple.
The changed RDF graphs will still be
OWL 2 DL ontologies in RDF graph form,
since annotations are strictly optional in OWL 2 DL ontologies
and may therefore be omitted.
Also, this operation will not change the formal meaning of the ontologies
under the OWL 2 Direct Semantics,
since annotations do not have a formal meaning under this semantics.

Reason 2: An Entity Declaration exclusively in F(G2*).
The ontology F(G2*)
contains an entity declaration for the class IRI
ex:c3 (4),
for which there is no corresponding entity declaration
in F(G1*).
The OWL 2 Direct Semantics does not give a formal meaning to
entity declarations,
while the OWL 2 RDF-Based Semantics gives a formal meaning
to the corresponding declaration statement (4) in G2*.
The consequences are analog to those described for reason 1.

Resolution of Reason 2.
The declaration statement (4) in G2*
will be copied to G1*.
An OWL 2 RDF-Based interpretation
that satisfies the modified graph G1*
will then also satisfy the declaration statement.
The changed RDF graphs will still be
OWL 2 DL ontologies in RDF graph form,
since the copied declaration statement is not in conflict
with any of the other entity declarations
in G1*.
Also, this operation will not change the formal meaning of the ontologies
under the OWL 2 Direct Semantics,
since entity declarations do not have a formal meaning under this semantics.

Reason 3: Different Ontology IRIs in F(G1*) and F(G2*).
The ontology IRIs for the two ontologies,
given by (1) in F(G1*)
and by (1) in F(G2*),
differ from each other.
The OWL 2 Direct Semantics does not give a formal meaning to
ontology headers,
while the OWL 2 RDF-Based Semantics gives a formal meaning
to the corresponding header triples
(1) in G1*
and
(1) in G2*.
Since these header triples differ from each other,
the consequences are analog to those described for reason 1.

Resolution of Reason 3.
The IRI
in the subject position of the header triple (1)
in G2*
is changed into a blank node.
Due to the existential semantics of blank nodes under the OWL 2 RDF-Based Semantics
the resulting triple will then be entailed
by triple (1)
in G1*.
The changed RDF graphs will still be
OWL 2 DL ontologies in RDF graph form,
since an ontology IRI is optional for an OWL 2 DL ontology.
(Note, however, that it would have been an error to simply remove
triple (1) from G2*,
since an OWL 2 DL ontology is required to contain an ontology header.)
Also, this operation will not change the formal meaning of the ontologies
under the OWL 2 Direct Semantics,
since ontology headers do not have a formal meaning under this semantics.

Reason 4: A Class Expression in F(G2*).
Axiom (5) of F(G2*)
contains a class expression
that represents the union of the two classes
denoted by
ex:c2
and
ex:c3.
Within G2*,
this class expression is represented
by the triples (6) and (7),
both having the blank node
"_:x"
in their respective subject position.
The way the OWL 2 RDF-Based Semantics interprets these two triples
differs from the way
the OWL 2 Direct Semantics treats the class expression
in axiom (5) of F(G2*).

The OWL 2 Direct Semantics treats classes as sets,
i.e. subsets of the universe.
Thus,
the IRIs
ex:c2
and
ex:c3
in F(G2*)
denote two sets,
and the class expression
in axiom (5) of F(G2*)
therefore represents the set
that consists of the union of these two sets.

The OWL 2 RDF-Based Semantics,
on the other hand,
treats classes as individuals,
i.e. members of the universe.
While every class under the OWL 2 RDF-Based Semantics
represents a certain subset of the universe,
namely its class extension,
this set is actually distinguished from the class itself.
For two given classes
it is ensured under the OWL 2 RDF-Based Semantics,
just as for the OWL 2 Direct Semantics,
that the union of their class extensions will always exist
as a subset of the universe.
However,
there is no guarantee
that there will also exist
an individual in the universe
that has this set union as its class extension.

Under the OWL 2 RDF-Based Semantics,
triple (7) of G2*
essentially claims that a class exists
being the union of two other classes.
But since
the existence of such a union class
is not ensured by G1*,
there will be OWL 2 RDF-Based interpretations
that satisfy G1*
without satisfying
triple (7) of G2*.
Hence,
G1*
does not
OWL 2 RDF-Based entail
G2*.

Resolution of Reason 4.
The triples (6) and (7) of G2*
are copied to G1*
together with the new triple
"_:x owl:equivalentClass _:x".
In addition,
for the IRI
ex:c3,
which only occurs in the union class expression
but not in G1*,
an entity declaration is added
to G1*
by the resolution of reason 2.
If an OWL 2 RDF-Based interpretation satisfies the modified graph G1*,
then the triples (6) and (7) of G2*
will now be satisfied.
The changed RDF graphs will still be
OWL 2 DL ontologies in RDF graph form,
since the whole set of added triples
validly encodes an OWL 2 axiom,
and since none of the restrictions on OWL 2 DL ontologies is hurt.
Also, this operation will not change
the formal meaning of the ontologies
under the OWL 2 Direct Semantics,
since the added equivalence axiom
is a tautology under this semantics.

Note that it would have been an error
to simply copy the
triples (6) and (7) of G2*
to G1*,
without also adding the new triple
"_:x owl:equivalentClass _:x".
This would have produced a class expression
that has no connection to any axiom in the ontology.
An OWL 2 DL ontology is basically a set of axioms
and does not allow for the occurrence of
"dangling" class expressions.
This is the reason for actually "embedding" the class expression
in an axiom.
It would have also been wrong
to use an arbitrary axiom for such an embedding,
since it has to be ensured
that the formal meaning of the original ontology does not change
under the OWL 2 Direct Semantics.
However,
any tautological axiom
that contains the original class expression
would have been sufficient for this purpose as well.

Complete Resolution: The Transformed Entailment Query.

Combining the resolutions of all the above reasons
leads to the following new pair of RDF graphs
( G1 , G2 ):

The following list reiterates the changes compared to the original RDF graphs
G1* and G2*:

Resolution of Reason 1 (Annotation): Triple (8) in G2* has been removed, i.e. there is no corresponding annotation triple in G2.

Resolution of Reason 2 (Entity Declaration): Triple (4) in G2* has been copied to G1*, becoming triple (4) in G1.

Resolution of Reason 3 (Ontology IRIs): The IRI in the subject position of triple (1) in G2* has been changed into a blank node, becoming triple (1) in G2.

Resolution of Reason 4 (Class Expression): Triples (6) and (7) in G2* have been copied to G1* together with the new triple "_:x owl:equivalentClass _:x", becoming triples (6), (7) and (8) in G1.

G1 and G2
are again
OWL 2 DL ontologies in RDF graph form
and can be mapped to the following
OWL 2 DL ontologies in Functional Syntax form
F(G1) and F(G2),
which again mutually meet the restrictions on OWL 2 DL ontologies:

As said earlier,
all the applied changes
preserve the formal meaning
of the original OWL 2 DL ontologies
under the OWL 2 Direct Semantics.
Hence,
it is still the case
that
F(G1)
OWL 2 Direct entails
F(G2).
However,
due to the syntactic transformation
the situation has changed for the OWL 2 RDF-Based Semantics:
it is now possible to show,
by following the lines of argumentation
for the resolutions of the different reasons given above,
that G1 OWL 2 RDF-Based entails G2
as well.

7.2 Correspondence Theorem

This section presents the OWL 2 correspondence theorem,
which compares the semantic expressivity of
the OWL 2 RDF-Based Semantics
with that of
the OWL 2 Direct Semantics.
The theorem basically states that
the OWL 2 RDF-Based Semantics is able to reflect all the semantic conclusions
of the OWL 2 Direct Semantics,
where the notion of a "semantic conclusion"
is technically expressed in terms of an entailment.

However,
as discussed in
Section 7.1,
there exist semantic differences
between the OWL 2 RDF-Based Semantics and the OWL 2 Direct Semantics,
which do not allow for stating
that any OWL 2 DL entailment query
that is an OWL 2 Direct entailment
will always also be an
OWL 2 RDF-Based entailment.
Nevertheless,
it can still be ensured that
any given OWL 2 DL entailment query
can be substituted
by another OWL 2 DL entailment query
in a way
that for the substitute entailment query
the desired relationship will really hold,
while preserving the formal meaning
compared to the original entailment query
under the OWL 2 Direct Semantics.

In fact,
the theorem only makes the seemingly weak assertion
that such a substitute entailment query
will always exist.
But the actual
proof for the theorem
in Section 7.3
will be more concrete
in that it will substitute each given OWL 2 DL entailment query
with a variant
that can be algorithmically constructed
by applying a set of simple syntactic transformations
to the original entailment query.
One can get an idea of how this works
from Section 7.1.

Technical Note on Corresponding Datatype Maps.
A distinction exists
between the format of
an OWL 2 RDF-Based datatype map,
as defined by Definition 4.1,
and the format of an OWL 2 Direct datatype map,
as defined in
Section 2.1 of the OWL 2 Direct Semantics
[OWL 2 Direct Semantics].
It is, however, possible to translate
between
an OWL 2 RDF-Based datatype map D
and
the corresponding OWL 2 Direct datatype map F(D)
in the following way:

Then,
there exist RDF graphs
G1 and G2
that are
OWL 2 DL ontologies in RDF graph form,
such that all the following relationships hold,
with
F(G1) and F(G2)
being the
OWL 2 DL ontologies in Functional Syntax form
that result from applying the reverse RDF mapping
to
G1 and G2,
respectively:

7.3 Proof for the Correspondence Theorem

This is the sketch of a proof for
Theorem 7.1 (OWL 2 Correspondence Theorem)
in
Section 7.2.
The proof sketch provides
the basic line of argumentation for showing the theorem.
However,
for complexity reasons,
some technical aspects of the theorem are only coarsely treated,
and the proof sketch also refrains
from considering the full amount of OWL 2 language constructs.
For certain steps of the proof
there are example calculations
that focus only on a small fraction of language constructs,
but which can be taken as a hint
on how a complete proof
taking into account every feature of the OWL 2 RDF-Based Semantics
could be constructed in principle.
A complete proof could make use of the observation
that the definitions of the OWL 2 Direct Semantics
and the OWL 2 RDF-Based Semantics,
despite their technical differences
as outlined in Section 7.1,
are closely aligned with respect to the different language constructs of OWL 2.

The proof sketch will make use of an approach
that will be called "balancing" throughout this section,
and which will now be introduced.
The basic idea is to substitute
the original pair of RDF graphs in an OWL 2 DL entailment query
by another entailment query
having the same semantic characteristics
under the OWL 2 Direct Semantics,
but for which the technical differences
between the two semantics specifications
have no relevant consequences
under the OWL 2 RDF-Based Semantics anymore.
A concrete example
for the application of this approach
was given in Section 7.1.

Definition (Balanced):
A pair of RDF graphs
( G1 , G2 )
is called
balanced,
if and only if
G1 and G2
are OWL 2 DL ontologies in RDF graph form,
such that all the following conditions hold,
with
F(G1) and F(G2)
being the
OWL 2 DL ontologies in
Functional Syntax
form
[OWL 2 Specification]
that result from applying the
reverse RDF mapping
[OWL 2 RDF Mapping]
to
G1 and G2,
respectively:

G2 contains exactly one ontology header consisting of a single RDF triple of the form "xrdf:type owl:Ontology", where x is either a blank node or, if an ontology IRI is used in G1, may alternatively equal that ontology IRI (see Table 4 in the OWL 2 RDF Mapping [OWL 2 RDF Mapping]).

Let the graph pair
( G1* , G2* )
be a valid input.
The resulting RDF graphs
G1 and G2
are constructed as follows,
starting from copies of
G1* and G2*,
respectively.

Since the initial versions of
G1 and G2
are OWL 2 DL ontologies in RDF graph form,
the canonical parsing process (CP)
for computing the reverse RDF mapping,
as described in
Section 3 of the OWL 2 RDF Mapping
[OWL 2 RDF Mapping],
can be applied.
Based on CP, it is possible to identify within these graphs

Consistently substitute all blank nodes in G2 such that G1 and G2 have no common blank nodes.

Apply CP to G1 and G2 (without changing these graphs) to identify the entity types of the IRIs, the anonymous individuals, and the sub graphs encoding OWL 2 language constructs.

For each sub graph g of G2: remove g from G2, if g is the RDF encoding of

an annotation, or

a deprecation statement, or

an annotation property axiom.

For the sub graph g of G2 corresponding to the ontology header in F(G2): substitute g in G2 by a triple of the form "xrdf:type owl:Ontology", where x is a new blank node not yet used in G2.

For each non-built-in IRI u in G1 and G2 and for each entity type T of u identified by CP: add to G1 or G2, respectively, the RDF triple "urdf:typet", where t is the vocabulary class IRI corresponding to T.

For each plain or typed literal L in G2: add to G1 the RDF triple "ordfs:commentL", where o is the IRI or blank node of the ontology header triple "ordf:typeowl:Ontology" in G1.

For each sub graph g of G2 that is the RDF encoding of an entity declaration: add g to G1.

For each sub graph g of G2 that is the RDF encoding of a property expression with root blank node x: add g to G1 together with the RDF triple "xowl:equivalentPropertyx".

For each sub graph g of G2 that is the RDF encoding of a class expression with root blank node x: add g to G1 together with the RDF triple "xowl:equivalentClassx".

For each sub graph g of G2 that is the RDF encoding of a data range expression with root blank node x:

If g is part of a data property restriction expression, then nothing needs to be done, since the comprising restriction expression is covered by the treatment of class expressions, and therefore g occurs in G1 as well.

Otherwise, add a declaration triple to G1 for a new data property p that does not yet occur in G1 and G2. Then, the RDF encoding r of a universal data property restriction expression on property p is created for g. Let r have the new root blank node y. Add r to G1 together with the RDF triple "yowl:equivalentClassy".

For each sub graph g of G2 that is an RDF sequence with root blank node x, which does not occur in the RDF encoding of language constructs already treated by one of the earlier steps, i.e. g is part of the encoding of an axiom: create the RDF encoding r of an enumeration class expression with a new root blank node y having the main RDF triple "yowl:oneOfx". Then, add r to G1 together with the RDF triple "yowl:equivalentClassy". Additionally, for every IRI u being a member of the RDF sequence, add to G1 a typing triple "urdf:type owl:NamedIndividual". If one of the sequence members is a blank node z that is the root node of some property expression or class expression e, then select a new IRI w not yet occurring in G1, consistently replace z by w everywhere in r, add to G1 the triple "wowl:equivalentPropertyz" or "wowl:equivalentClassz", respectively, and add to G1 the two triples "wrdf:type owl:NamedIndividual" and "wrdf:typet", where t is the vocabulary class IRI that represents the appropriate entity type of the expression e. No further treatment of e is needed, since e is treated by the earlier steps covering expressions.

In the following it is shown that all the claims of the balancing lemma hold.

A: Existence of a Terminating Algorithm.
An algorithm exists
for mapping
the input graph pair
( G1* , G2* )
to the output graph pair
( G1 , G2 ),
since CP (applied in step 2)
is described in the form of an algorithm
in the
OWL 2 RDF Mapping
[OWL 2 RDF Mapping],
and since all other steps
can obviously be performed algorithmically.
The algorithm terminates,
since CP terminates
on arbitrary input graphs,
and since all other steps
can obviously be executed in finite time.

B: The Resulting RDF Graphs are OWL 2 DL Ontologies.
The RDF graphs
G1 and G2
are OWL 2 DL ontologies in RDF graph form
that mutually meet
the restrictions on OWL 2 DL ontologies,
since the original RDF graphs
G1* and G2*
have this feature,
and since each of the steps described above
transforms a pair of RDF graphs with this feature
again into a pair of RDF graphs with this feature,
for the following reasons:

The consistent substitution of blank nodes in step 1 does not change the structure of an OWL 2 DL ontology.

The application of CP in step 2 does not change the graphs.

Annotations, deprecation statements and annotation property axioms are optional information in an OWL 2 DL ontology and can therefore be omitted in step 3.

The ontology header of an OWL 2 DL ontology does neither require the existence of an ontology IRI nor of any ontology properties, and so the substitution of the ontology header in step 4 is a valid operation.

If an entity has some particular entity type for which there is no explicitly given entity declaration, then the entity declaration may be added, as done in step 5.

It is allowed to add arbitrary annotations to the ontology header of an OWL 2 DL ontology, as done in step 6.

Entity declarations may be copied from G2 to G1 in step 7 without conflict, since the original ontologies have been assumed to mutually meet the restrictions on OWL 2 DL ontologies regarding different entity declarations for the same IRI (e.g. that one IRI must not be the name of both an object property and a data property).

Adding to G1 an axiom that claims equivalence of some property expression (step 8) or class expression (step 9) with itself, where the expression already occurs in G2, is an allowed operation, since the original ontologies are assumed to mutually meet the restrictions on OWL 2 DL ontologies concerning property and class expressions, and since no syntactic restrictions exist on this specific use of equivalence axioms.

For the case of data ranges (step 10) it is sufficient to note that placing universal property restrictions on arbitrary (simple or complex) property expressions is allowed in OWL 2 DL. The rest of the argumentation follows the lines of the treatment of class expressions in step 9.

For the treatment of RDF sequences in step 11: First, the enumeration class expressions being constructed from the RDF sequences are syntactically valid in OWL 2 DL, since all enumerated entries are IRIs by construction. Second, there is no restriction in OWL 2 DL disallowing axioms that claim equivalence of enumeration class expressions with themselves. Third, punning in OWL 2 DL allows a given non-built-in IRI of any entity type to be additionally declared as a named individual. Forth, there is no OWL 2 DL restriction forbidding to add an entity declaration for a new (i.e. not elsewhere used) IRI and to assert the denotation of this new IRI to be equivalent to some existing property or class expression. Hence, the resulting ontologies still mutually meet all syntactic restrictions on OWL 2 DL ontologies.

C: The Resulting Pair of RDF Graphs is Balanced.
All the conditions of balanced pairs of RDF graphs
are met by the pair
( G1 , G2 )
for the following reasons:

Condition 1: It has already been shown in paragraph B that G1 and G2 mutually meet the restrictions on OWL 2 DL ontologies.

Conditions 2.1 and 2.2 on nodes in G1 and G2 are met by steps 5 and 6, respectively.

Conditions 4.1, 4.3 and 4.4 on annotations, deprecation statements and annotation property axioms in G2, respectively, are all satisfied by step 3.

Condition 4.2 on statements with ontology properties is implicitly satisfied by step 4, since the substitution of the ontology header in G2 removes all existing statements with ontology properties.

Condition 5.1 on entity declarations in G2 being reflected in G1 is satisfied by step 7.

Conditions 5.2, 5.3 and 5.4 on property, class and data range expressions in G2, respectively, being reflected in G1 are met by steps 8, 9 and 10, respectively.

Condition 5.5 on RDF sequences in G2 being reflected in G1 is satisfied by step 11.

D: The Resulting Ontologies are semantically equivalent with the Original Ontologies under the OWL 2 Direct Semantics.
F(G1) is semantically equivalent with F(G1*),
since F(G1) differs from F(G1*) only by (potentially):

additional entity declarations (steps 5, 7 and 11), which have no formal meaning under the OWL 2 Direct Semantics;

additional annotations (step 6), which have no formal meaning;

additional tautological axioms (steps 8, 9, 10 and 11), which do not change the formal meaning;

F(G2) is semantically equivalent with F(G2*),
since F(G2) differs from F(G2*) only by (potentially):

additional entity declarations (step 5), which have no formal meaning.

End of Proof for the Balancing Lemma.

In the following,
the correspondence theorem will be proven.

Assume that the premises of the correspondence theorem are true
for a given pair
( G1* , G2* )
of RDF graphs.
This allows for applying the
balancing lemma,
which provides the existence of corresponding RDF graphs
G1 and G2
that are OWL 2 DL ontologies in RDF graph form,
and which meet the
definition of balanced graph pairs.
Let F(G1) and F(G2)
be the corresponding OWL 2 DL ontologies in Functional Syntax form.
Then,
the claimed relationship 1 of the correspondence theorem
follows directly from relationship 1 of the balancing lemma
and from condition 1 of the definition of balanced graph pairs.
Further,
the claimed relationships 2 and 3 of the correspondence theorem
follow directly from the relationships 2 and 3 of the balancing lemma,
respectively.

The rest of this proof will treat
the claimed relationship 4 of the correspondence theorem,
which states that
if F(G1) OWL 2 Direct entails F(G2)
with respect to F(D),
then G1 OWL 2 RDF-Based entails G2
with respect to D.
For this to see,
an arbitrary OWL 2 RDF-Based interpretation I will be selected
that OWL 2 RDF-Based satisfies G1.
For I,
a closely corresponding OWL 2 Direct interpretation J
will be constructed,
and it will then be shown
that J OWL 2 Direct satisfies F(G1).
Since it was assumed that
F(G1) OWL 2 Direct entails F(G2),
it will follow that J OWL 2 Direct satisfies F(G2).
Based on this result, it will then be possible to show
that I also OWL 2 RDF-Based satisfies G2.
Since I was arbitrarily selected,
this will mean
that G1 OWL 2 RDF-Based entails G2.

Since the pair
( G1 , G2 )
is balanced,
there exist entity declarations
in F(G1)
for each entity type
of every non-built-in IRI
occurring in G1:
For each entity declaration
of the form
"Declaration(T(u))"
in F(G1),
such that T is the entity type for some IRI u,
a typing triple
of the form
"urdf:typet"
exists in G1,
where t is the vocabulary class IRI
representing the part of the universe of I
that corresponds to T.
Since I OWL 2 RDF-Based satisfies G1,
all these declaration typing triples are OWL 2 RDF-Based satisfied by I,
and thus all non-built-in IRIs in G1
are instances of all their declared parts of the universe of I.

If u is declared as a class, then set uC := ICEXT(I(u)), since G1 contains the triple "urdf:type owl:Class", i.e., I(u) ∈ IC.

If u is declared as an object property, then set uOP := IEXT(I(u)), since G1 contains the triple "urdf:type owl:ObjectProperty", i.e., I(u) ∈ IP.

If u is declared as a data property, then set uDP := IEXT(I(u)), since G1 contains the triple "urdf:type owl:DatatypeProperty", i.e., I(u) ∈ IODP.

If u is declared as a named individual, then set uI := I(u), since G1 contains the triple "urdf:type owl:NamedIndividual", i.e., I(u) ∈ IR.

If u is declared as a datatype, then set uDT := ICEXT(I(u)), since G1 contains the triple "urdf:type rdfs:Datatype", i.e., I(u) ∈ IDC.

Notes:

A literal occurring in G1 is mapped by the reverse RDF mapping to the same literal in F(G1), and the formal meaning of a well-formed literal is analog for both the OWL 2 RDF-Based Semantics and the OWL 2 Direct Semantics.

A blank nodeb occurring in G1 that represents an anonymous individual is written as the same blank node b in F(G1). Both the OWL 2 RDF-Based Semantics and the OWL 2 Direct Semantics treat anonymous individuals in an analog way as existential variables defined locally to a given ontology, i.e. some individual x exists in the universe to which all occurrences of b in the ontology can be mapped (see Section 1.5 in the RDF Semantics [RDF Semantics] for the precise definition on how blank nodes are treated under the OWL 2 RDF-Based Semantics). Hence, the same mapping from b to x can be used with both I and J.

G1 may also contain declarations for annotation properties. Since annotation properties have no formal meaning under the OWL 2 Direct Semantics, the OWL 2 Direct interpretation J does not treat them.

With the above definition it is possible for J to have a nonseparated vocabulary according to Section 5.9 of the OWL 2 Structural Specification [OWL 2 Specification]. Since G1 is an OWL 2 DL ontology in RDF graph form, it is allowed that the same IRI u may be declared as one or more of an individual name, either a class name or a datatype name, and either an object property name or a data property name. For the OWL 2 RDF-Based interpretation I, the IRI u will always denote the same individual in the universe IR, where I(u) may additionally have a class extension or a property extension, or both. For the OWL 2 Direct interpretation J, however, u will denote as an individual name an element of ΔI, as a class name a subset of ΔI, as a datatype name a subset of ΔD, as an object property name a subset of ΔI × ΔI, and as a data property name a subset of ΔI × ΔD.

Step 2: Satisfaction of F(G1) by the OWL 2 Direct Interpretation.

Based on the premise that I OWL 2 RDF-Based satisfies G1,
it has to be shown that J OWL 2 Direct satisfies F(G1).
For this to hold,
it will be sufficient that
J OWL 2 Direct satisfies every axiom A occurring in F(G1).
Let gA be the sub graph of G1
that is mapped to A by the reverse RDF mapping.
The basic idea can roughly be described as follows:

Since I is an OWL 2 RDF-Based interpretation,
all the OWL 2 RDF-Based semantic conditions are met by I.
Due to the close alignment between the definitions
in the OWL 2 RDF-Based Semantics
and the OWL 2 Direct Semantics,
OWL 2 RDF-Based semantic conditions exist
that semantically correspond
to the definition of the interpretation of the axiom A.
In particular,
the antecedent of one of these semantic conditions
will become true,
if the RDF-encoding of A,
i.e. the graph gA,
is satisfied
(in the case of an "if-and-only-if" semantic condition
this will generally be the left-to-right direction of that condition).
Now,
all the RDF triples in gA
are OWL 2 RDF-Based satisfied by I,
since I OWL 2 RDF-Based satisfies G1.
Hence,
the antecedent of the semantic condition becomes true,
and therefore its consequent becomes true as well.
This will reveal a certain semantic relationship
that
according to I
holds between the denotations of the
IRIs, literals and anonymous individuals
occurring in gA,
which,
roughly speaking,
expresses the meaning of the OWL 2 axiom A.
Because of the close semantic correspondence
of the OWL 2 Direct interpretation J to I,
the analog semantic relationship holds
according to J
between the denotations of the
IRIs, literals and anonymous individuals
occurring in A.
This semantic relationship
turns out to be compatible
with the formal meaning of the axiom A
as specified by the OWL 2 Direct Semantics,
i.e. J satisfies A.

This basic idea is now demonstrated in more detail
for a single example axiom A in F(G1),
which can be taken as a hint on
how a complete proof
taking into account every feature of the OWL 2 RDF-Based Semantics
could be constructed in principle.

Let A be the following OWL 2 axiom in F(G1):

A : SubClassOf(ex:c1 ObjectUnionOf(ex:c2 ex:c3))

and let gA be the corresponding sub graph in G1
that is being mapped to A via the reverse RDF mapping,
namely

that correspond to class entity declarations in F(G1) for the IRIs
"ex:c1",
"ex:c2", and
"ex:c3",
respectively.
All these declaration typing triples are OWL 2 RDF-Based satisfied by I,
since it has been postulated
that I OWL 2 RDF-Based satisfies G1.
Hence,
by applying the semantics of rdf:type
(see
Section 4.1 of the RDF Semantics
[RDF Semantics]),
all the IRIs denote classes, precisely:

I(ex:c1) ∈ IC ,I(ex:c2) ∈ IC , andI(ex:c3) ∈ IC .

Since I is an OWL 2 RDF-Based interpretation,
it meets all the OWL 2 RDF-Based semantic conditions,
and since I OWL 2 RDF-Based satisfies G1,
all the triples in gA are OWL 2 RDF-Based satisfied.
This meets the left-to-right directions of the semantic conditions
for subclass axioms
("rdfs:subClassOf",
see Section 5.8)
and union class expressions
("owl:unionOf",
see Section 5.4),
which results in the following semantic relationship
that holds between the extensions of the classes above
according to I:

ICEXT(I(ex:c1))
⊆
ICEXT(I(ex:c2))
∪
ICEXT(I(ex:c3)) .

By applying the definition of J,
one can conclude
that the following semantic relationship
holds between the denotations of the class names occurring in A
according to J:

(ex:c1) C
⊆
(ex:c2) C
∪
(ex:c3) C .

This semantic relationship is compatible
with the formal meaning of the axiom A
under the OWL 2 Direct Semantics.
Hence, J OWL 2 Direct satisfies A.

Since J OWL 2 Direct satisfies F(G1),
and since it has been postulated that
F(G1) OWL 2 Direct entails F(G2),
it follows that
J OWL 2 Direct satisfies F(G2).

Step 3: Satisfaction of G2 by the OWL 2 RDF-Based Interpretation.

The last step will be
to show that I OWL 2 RDF-Based satisfies G2.
For this to hold,
I needs to OWL 2 RDF-Based satisfy
every triple occurring in G2.
The basic idea can roughly be described as follows:

First:
According to the "semantic conditions for ground graphs"
in Section 1.4 of the RDF Semantics specification
[RDF Semantics],
all the IRIs and literals used in RDF triples in G2
need to be in the vocabulary VI of I.
This is true for the following reason:
Since the pair
( G1 , G2 )
is balanced,
all IRIs and literals occurring in G2
do also occur in G1.
Since I satisfies G1,
all IRIs and literals in G1,
including those in G2,
are contained in VI
due to the semantic conditions for ground graphs.

Second:
If a set of RDF triples encodes an OWL 2 language construct
that is not interpreted by the OWL 2 Direct Semantics,
such as annotations,
then G2 should contain such a set of RDF triples
only if they are also included in G1.
The reason is
that with such triples
there will, in general, exist OWL 2 RDF-Based interpretations
only satisfying the graph G1 but not G2,
which will render the pair
( G1 , G2 )
into a nonentailment
(an exception are RDF triples
that are true
under every OWL 2 RDF-Based interpretation).
Since the pair
( G1 , G2 )
is balanced,
G2 will not contain the RDF encoding for any
annotations,
statements with ontology properties,
deprecation statements or
annotation property axioms.
Hence,
there are no corresponding RDF triples that need to be satisfied by I.

Third:
Since G2 is an OWL 2 DL ontology in RDF graph form,
the graph is partitioned by the
reverse RDF mapping
[OWL 2 RDF Mapping]
into sub graphs corresponding to
either ontology headers,
entity declarations
or axioms,
where axioms may further consist of different kinds of expressions,
such as Boolean class expressions.
It has to be shown that all the triples in each such sub graph
are OWL 2 RDF-Based satisfied by I.

For ontology headers:
Let A be the ontology header of F(G2)
and let gA be the corresponding sub graph of G2.
Since the pair
( G1 , G2 )
is balanced,
gA is encoded as a single RDF triple of the form
"xrdf:type owl:Ontology",
where x is either an IRI or a blank node.
Since G1 is an OWL 2 DL ontology in RDF graph form,
G1 also contains the encoding of an ontology header
including a triple g1 of the form
"yrdf:type owl:Ontology",
where y is either an IRI or a blank node.
Since I OWL 2 RDF-Based satisfies G1,
g1 is satisfied by I.
If both y and x are IRIs,
then, due to balancing,
x equals y,
and therefore gA equals g1,
i.e. gA is OWL 2 RDF-Based satisfied by I.
Otherwise,
balancing forces x to be a blank node,
i.e. x is treated as an existential variable
under the OWL 2 RDF-Based Semantics
according to the
"semantic conditions for blank nodes"
[RDF Semantics].
From this observation,
and from the premise that I satisfies g1,
it follows that gA is OWL 2 RDF-Based satisfied by I.

For entity declarations:
Let A be an entity declaration in F(G2),
and let gA be the corresponding sub graph of G2.
Since the pair
( G1 , G2 )
is balanced,
A occurs in F(G1),
and hence gA is a sub graph of G1.
Since I OWL 2 RDF-Based satisfies G1,
I OWL 2 RDF-Based satisfies gA.

For axioms:
Let A be an axiom in F(G2),
and let gA be the corresponding sub graph of G2.
Since I is an OWL 2 RDF-Based interpretation,
all the OWL 2 RDF-Based semantic conditions are met by I.
Due to the close alignment between the definitions
in the OWL 2 RDF-Based Semantics
and the OWL 2 Direct Semantics,
OWL 2 RDF-Based semantic conditions exist
that semantically correspond
to the definition of the interpretation of the axiom A.
In particular,
the consequent of one of these semantic conditions
corresponds to the RDF-encoding of A,
i.e. the graph gA,
except for declaration typing triples,
for which satisfaction has already been shown
(in the case of an "if-and-only-if" semantic condition
this will generally be the right-to-left direction of that condition).
Hence,
in order to show that gA is OWL 2 RDF-Based satisfied by I,
it will be sufficient to show
that the antecedent of this semantic condition is true.
In general,
several requirements have to be met to ensure this:

Requirement 1:
The denotations of all the non-built-in IRIs in gA
have to be contained in the appropriate part of the universe of I.
This can be shown as follows.
For every non-built-in IRI u occurring in gA,
u also occurs in A.
Since the pair
( G1 , G2 )
is balanced,
there are entity declarations in F(G2)
for all the entity types of u,
each being of the form
D := "Declaration(T(u))"
for some entity type T.
From the reverse RDF mapping follows
that for each such declaration D
a typing triple d exists in G2,
being of the form d := "urdf:typet",
where t is the vocabulary class IRI
representing the part of the universe of I
that corresponds to the entity type T.
It has already been shown that,
for D being an entity declaration in F(G2)
and d being the corresponding sub graph in G2,
I OWL 2 RDF-Based satisfies d.
Hence, I(u) is an individual
contained in the appropriate part of the universe.

Requirement 2:
For every expression E occurring in A,
with the RDF encoding gE in gA,
an individual has to exist in the universe of I
that appropriately represents the denotation of E.
Since I is an OWL 2 RDF-Based interpretation,
all the OWL 2 RDF-Based semantic conditions are met by I.
Due to the close alignment between the definitions
in the OWL 2 RDF-Based Semantics
and the OWL 2 Direct Semantics,
OWL 2 RDF-Based semantic conditions exist
that semantically correspond
to the definition of the interpretation of the expression E.
In particular,
the antecedent of one of these semantic conditions
will become true,
if the RDF-encoding of E,
i.e. the graph gE,
is satisfied
(in the case of an "if-and-only-if" semantic condition
this will generally be the left-to-right direction of that condition).
Now,
since the pair
( G1 , G2 )
is balanced,
gE also occurs in G1.
So,
since I OWL 2 RDF-Based satisfies G1,
gE is OWL 2 RDF-Based satisfied by I.
Hence,
the antecedent of the semantic condition becomes true,
and therefore its consequent becomes true as well.
This will result in the existence of an individual with the required properties,
when taking into account existential blank node semantics.

Requirement 3:
A semantic relationship
has to hold
between the denotations of the
IRIs, literals and anonymous individuals
occurring in gA
with respect to I,
which,
roughly speaking,
expresses the meaning of the OWL 2 axiom A.
This is the case for the following reasons:
First,
the literals and anonymous individuals
occurring in A and gA, respectively,
are interpreted in an analog way
under the OWL 2 Direct Semantics and the OWL 2 RDF-Based Semantics.
Second,
it was assumed that the OWL 2 Direct interpretation J
OWL 2 Direct satisfies A,
and therefore a semantic relationship
with the desired properties
holds with respect to J.
Third,
J has been defined in close correspondence to I,
so that for the semantic relationship expressed by J
an analog semantic relationship holds with respect to I.

This basic idea is now demonstrated in more detail
for a single example axiom A in F(G2),
which can be taken as a hint on
how a complete proof
taking into account every feature of the OWL 2 RDF-Based Semantics
could be constructed in principle.

Let A be the following OWL 2 axiom in F(G2):

A : SubClassOf(ex:c1 ObjectUnionOf(ex:c2 ex:c3))

and let gA be the corresponding sub graph in G2
that is being mapped to A via the reverse RDF mapping,
namely

that correspond to class entity declarations in F(G2) for the IRIs
"ex:c1",
"ex:c2", and
"ex:c3",
respectively.
All these declaration typing triples are OWL 2 RDF-Based satisfied by I,
since due to balancing
the typing triples exist in G1 as well,
and since it has been postulated
that I OWL 2 RDF-Based satisfies all triples in G1.
Hence,
by applying the semantics of rdf:type
(see
Section 4.1 of the RDF Semantics
[RDF Semantics]),
all the IRIs denote classes,
and therefore the denotations of the IRIs
are included in the appropriate part of the universe of I,
precisely:

I(ex:c1) ∈ IC ,I(ex:c2) ∈ IC , andI(ex:c3) ∈ IC .

Second,
gA contains the sub graph gE,
given by

gE :

_:x rdf:type owl:Class ._:x owl:unionOf ( c2 c3 ) .

which corresponds to the union class expression E in A,
given by

E : ObjectUnionOf(ex:c2 ex:c3)

Since the pair
( G1 , G2 )
is balanced,
gE occurs as a sub graph of G1 as well.
gE contains blank nodes
and,
since I satisfies G1,
the semantic conditions for RDF graphs with blank nodes apply
(see
Section 1.5 of the RDF Semantics
[RDF Semantics]).
This provides the existence of
a mapping B from blank(gE) to IR,
where blank(gE) is
the set of all blank nodes occurring in gE.
It follows that
the extended interpretation I+B
OWL 2 RDF-Based satisfies all the triples in gE.
Further,
since I is an OWL 2 RDF-Based interpretation,
I meets all the OWL 2 RDF-Based semantic conditions.
Thus, the left-to-right direction
of the semantic condition for union class expressions
("owl:unionOf",
see Section 5.4)
applies, providing:

8 Appendix: Comprehension Conditions (Informative)

The correspondence theorem
in Section 7.2
shows
that it is possible for the OWL 2 RDF-Based Semantics
to reflect all the entailments of the
OWL 2 Direct Semantics
[OWL 2 Direct Semantics],
provided that one allows for certain "harmless" syntactic transformations
on the RDF graphs being considered.
This makes numerous potentially desirable and useful entailments available
that would otherwise be outside the scope of the OWL 2 RDF-Based Semantics,
for the technical reasons discussed in
Section 7.1.
It seems natural to ask for similar entailments
even
when an entailment query
does not consist of OWL 2 DL ontologies in RDF graph form.
However,
the correspondence theorem does not apply to such cases,
and thus the OWL 2 Direct Semantics cannot be taken
as a reference frame
for "desirable" and "useful" entailments,
or for when a graph transformation
can be considered "harmless" or not.

As discussed in
Section 7.1,
a core obstacle for the correspondence theorem to hold
was the RDF encoding of OWL 2 expressions,
such as union class expressions,
when they appear on the right hand side of an entailment query.
Under the OWL 2 RDF-Based Semantics
it is not generally ensured that an individual exists,
which represents the denotation of such an expression.
The "comprehension conditions" defined in this section
are additional semantic conditions
that provide the necessary individuals
for every sequence, class and property expression.
By this,
the combination
of the normative semantic conditions of the OWL 2 RDF-Based Semantics
(Section 5)
and the comprehension conditions
can be regarded to "simulate" the semantic expressivity
of the OWL 2 Direct Semantics
on entailment queries consisting of arbitrary RDF graphs.

The combined semantics is,
however,
not primarily intended for use in actual implementations.
The comprehension conditions add significantly
to the complexity and expressivity
of the basic semantics
and,
in fact,
have proven to
lead to formal inconsistency.
But
the combined semantics
can still be seen as a generalized reference frame
for "desirable" and "useful" entailments,
and this can be used,
for example,
to evaluate methods that syntactically transform unrestricted entailment queries
in order to receive additional entailments under the OWL 2 RDF-Based Semantics.
Such a concrete method is, however,
outside the scope of this specification.

8.1 Comprehension Conditions for Sequences

Table 8.1
lists the comprehension conditions for sequences,
i.e. RDF lists.
These comprehension conditions provide the existence
of sequences
built from any finite combination of individuals
contained in the universe.

8.2 Comprehension Conditions for Boolean Connectives

Table 8.2
lists the comprehension conditions for
Boolean connectives
(see Section 5.4
for the corresponding semantic conditions).
These comprehension conditions provide the existence
of complements for any class and datatype,
and of intersections and unions
built from any finite set of classes
contained in the universe.

Table 8.2: Comprehension Conditions for Boolean Connectives

if

then exists z ∈ IR

s sequence of c1 , … , cn ∈ IC

( z , s ) ∈ IEXT(I(owl:intersectionOf))

s sequence of c1 , … , cn ∈ IC

( z , s ) ∈ IEXT(I(owl:unionOf))

c ∈ IC

( z , c ) ∈ IEXT(I(owl:complementOf))

d ∈ IDC

( z , d ) ∈ IEXT(I(owl:datatypeComplementOf))

8.3 Comprehension Conditions for Enumerations

Table 8.3
lists the comprehension conditions for
enumerations
(see Section 5.5
for the corresponding semantic conditions).
These comprehension conditions provide the existence
of enumeration classes
built from any finite set of individuals
contained in the universe.

Table 8.3: Comprehension Conditions for Enumerations

if

then exists z ∈ IR

s sequence of a1 , … , an ∈ IR

( z , s ) ∈ IEXT(I(owl:oneOf))

8.4 Comprehension Conditions for Property Restrictions

Table 8.4
lists the comprehension conditions for
property restrictions
(see Section 5.6
for the corresponding semantic conditions).
These comprehension conditions provide the existence
of cardinality restrictions
on any property and for any nonnegative integer,
as well as value restrictions
on any property and on any class
contained in the universe.

Implementations are not required
to support the comprehension conditions for
owl:onProperties,
but
MAY
support them
in order to realize
n-ary dataranges with arity ≥ 2
(see
Sections
7
and
8.4
of the OWL 2 Structural Specification
[OWL 2 Specification]
for further information).

8.5 Comprehension Conditions for Datatype Restrictions

Table 8.5
lists the comprehension conditions for
datatype restrictions
(see Section 5.7
for the corresponding semantic conditions).
These comprehension conditions provide the existence
of datatypes
built from restricting any datatype
contained in the universe
by any finite set of facet-value pairs
contained in the facet space
(see Section 4.1)
of the original datatype.

Note that,
to some extent,
the OWL 2 RDF-Based Semantics already covers the use of inverse property expressions
by means of the semantic conditions of inverse property axioms
(see Section 5.12),
since these semantic conditions also apply to an existential variable
on the left hand side of an inverse property axiom.
Nevertheless,
not all relevant cases are covered by this semantic condition.
For example,
one might expect the above example graph
to be generally true.
However,
the normative semantic conditions
do not permit this conclusion,
since it is not ensured that
for every property p
there is an individual in the universe
with a property extension being inverse to that of p.

Table 8.6: Comprehension Conditions for Inverse Properties

if

then exists z ∈ IR

p ∈ IP

( z , p ) ∈ IEXT(I(owl:inverseOf))

9 Appendix: Changes from OWL 1 (Informative)

This section lists relevant differences
between the OWL 2 RDF-Based Semantics and the original specification of the
OWL 1 RDF-Compatible Semantics
[OWL 1 RDF-Compatible Semantics].
Significant effort has been spent
in keeping the design of the OWL 2 RDF-Based Semantics
as close as possible
to that of the OWL 1 RDF-Compatible Semantics.
While this aim was achieved to a large degree,
the OWL 2 RDF-Based Semantics actually deviates from its predecessor in several aspects.
In most cases this is because of serious technical problems
that would have arisen
from a conservative
semantic extension.
Not listed are
the new language constructs and the new datatypes of OWL 2.

The following markers are used:

[DEV]: a deviation from OWL 1 that breaks backward compatibility

[EXT]: a backward compatible extension to OWL 1

[NOM]: a change of the nomenclature originally used in OWL 1

[DPR]: a feature of OWL 1 that has been deprecated as of OWL 2

Generalized Graph Syntax [EXT].
The OWL 2 RDF-Based Semantics
allows RDF graphs to contain
IRIs
[RFC 3987]
(see Section 2.1),
whereas the OWL 1 RDF-Compatible Semantics was restricted to RDF graphs with
URIs
[RFC 2396].
This change is in accordance with the rest of the OWL 2 specification
(see
Section 2.4 of the OWL 2 Structural Specification
[OWL 2 Specification]).
In addition,
the OWL 2 RDF-Based Semantics
is now explicitly allowed to
be applied to RDF graphs containing
"generalized" RDF triples,
i.e. triples that can consist of
IRIs, literals or blank nodes
in all three positions
(Section 2.1),
although implementations are not required to support this.
In contrast,
the OWL 1 RDF-Compatible Semantics was restricted to RDF graphs
conforming to the
RDF Concepts specification
[RDF Concepts].
These limitations of the OWL 1 RDF-Compatible Semantics
were actually inherited from the
RDF Semantics specification
[RDF Semantics].
The relaxations are intended to warrant interoperability
with existing and future technologies and tools.
Both changes are compatible with OWL 1,
since all RDF graphs that were legal under the OWL 1 RDF-Compatible Semantics
are still legal under the OWL 2 RDF-Based Semantics.

Facets for Datatypes [EXT].
The basic definitions of a datatype and a D-interpretation,
as defined by the RDF Semantics specification
and as applied by the OWL 1 RDF-Compatible Semantics,
have been extended
to take into account constraining facets
(see Section 4),
in order to allow for datatype restrictions
as specified in Section 5.7.
This change is compatible with OWL 1,
since Section 5.1
of the RDF Semantics specification
explicitly allows for extending the minimal datatype definition provided there.

Correspondence Theorem and Comprehension Conditions [DEV].
The semantic conditions of the OWL 1 RDF-Compatible Semantics included
a set of so called
"comprehension conditions",
which allowed to prove the original
"correspondence theorem"
stating that every entailment of OWL 1 DL was also an entailment of OWL 1 Full.
The document at hand adds comprehension conditions
for the new language constructs of OWL 2
(see Section 8).
However,
the comprehension conditions
are not a normative aspect of the OWL 2 RDF-Based Semantics
anymore.
It has turned out
that combining the comprehension conditions
with the normative set of semantic conditions in
Section 5
would lead to formal inconsistency of the resulting semantics
(Issue 119).
In addition,
it became clear that
a correspondence theorem along the lines of the original theorem
would not work for the relationship between the
OWL 2 RDF-Based Semantics and the
OWL 2 Direct Semantics
[OWL 2 Direct Semantics],
since it is not possible to "balance" the differences between
the two semantics
solely by means of additional comprehension conditions
(see Section 7.1).
Consequently,
the correspondence theorem
of the OWL 2 RDF-Based Semantics
(Section 7.2)
follows an alternative approach
that replaces the use of the comprehension conditions
and can be seen as a technical refinement
of an idea
originally discussed by the WebOnt Working Group
(email).
This change is an incompatible deviation from OWL 1,
since certain aspects of the originally normative definition of the semantics
have been removed.

Flawed Semantics of Language Constructs with Argument Lists [DEV].
In the OWL 1 RDF-Compatible Semantics,
the semantic conditions for
unions, intersections and enumerations of classes
were defined in a flawed form,
which lead to formal inconsistency of the semantics
(Issue 120;
see also this
unofficial problem description).
The affected semantic conditions have been revised;
see
Section 5.4
and
Section 5.5.
This change is an incompatible deviation from OWL 1,
since the semantics has formally been weakened
in order to eliminate a source of inconsistency.

Incomplete Semantics of owl:AllDifferent [EXT].
The OWL 1 RDF-Compatible Semantics missed a certain semantic condition
for axioms based on the vocabulary term "owl:AllDifferent"
(see also this
unofficial problem description).
The missing semantic condition
has been added to the OWL 2 RDF-Based Semantics
(see Section 5.10).
This change is compatible with OWL 1,
since the semantics has been conservatively extended.

Aligned Semantics of owl:DataRange and rdfs:Datatype [EXT].
The class
owl:DataRange
has been made an equivalent class
to rdfs:Datatype
(see Section 5.2).
The main purpose for this change was
to allow for the deprecation of the term
owl:DataRange
in favor of rdfs:Datatype.
This change is compatible with OWL 1
according to an analysis
of the relationship between the two classes
in the OWL 1 RDF-Compatible Semantics
(email).

Ontology Properties as Annotation Properties [EXT].
Several properties
that have been ontology properties in OWL 1,
such as owl:priorVersion,
have now been specified
as both ontology properties and annotation properties,
in order to be in line
with the rest of the OWL 2 specification
(see
Section 5.5 of the OWL 2 Structural Specification
[OWL 2 Specification]).
This change is compatible with OWL 1,
since the semantics has been conservatively extended:
all the ontology properties of OWL 1 are still ontology properties in OWL 2.

Nonempty Data Value Enumerations [DEV].
The semantic condition for enumerations of data values
in Section 5.5
is now restricted to nonempty sets of data values.
This prevents the class owl:Nothing
from unintentionally becoming an instance
of the class rdfs:Datatype,
as analyzed in
(email).
This restriction of the semantics
is an incompatible deviation from OWL 1.
Note, however,
that it is still possible
to define a datatype as an empty enumeration of data values,
as explained in Section 5.5.

Terminological Clarifications [NOM].
This document uses the term "OWL 2 RDF-Based Semantics"
to refer to the specified semantics only.
According to Section 2.1,
the term "OWL 2 Full"
refers to the language
that is determined
by the set of all RDF graphs
(also called "OWL 2 Full ontologies")
being interpreted using the OWL 2 RDF-Based Semantics.
OWL 1 has not been particularly clear on this distinction.
Where the OWL 1 RDF-Compatible Semantics specification talked about
"OWL Full interpretations",
"OWL Full satisfaction",
"OWL Full consistency"
and
"OWL Full entailment",
the OWL 2 RDF-Based Semantics Specification talks
in Section 4
about
"OWL 2 RDF-Based interpretations",
"OWL 2 RDF-Based satisfaction",
"OWL 2 RDF-Based consistency"
and
"OWL 2 RDF-Based entailment",
respectively,
since these terms are primarily meant to be related to
the semantics
rather than the whole language.

Modified Abbreviations [NOM].
The names
"RI", "PI", "CI",
"EXTI", "CEXTI",
"SI", "LI" and "LVI",
which have been used in the
OWL 1 RDF-Compatible Semantics specification,
have been replaced by the corresponding names
defined in the
RDF Semantics document
[RDF Semantics],
namely "IR", "IP", "IC", "IEXT", "ICEXT", "IS", "IL" and "LV", respectively.
Furthermore,
all uses of the IRI mapping "IS"
have been replaced by the more general interpretation mapping "I",
following the conventions in the RDF Semantics document.
These changes are intended to support
the use of the OWL 2 RDF-Based Semantics document
as an incremental extension
of the RDF Semantics document.
Names for the "parts of the universe"
that were exclusively used in the OWL 1 RDF-Compatible Semantics document,
such as "IX" or "IODP",
have not been changed.
Other abbreviations,
such as "IAD" for the class extension of owl:AllDifferent,
have in general not been reused in the document at hand,
but the explicit nonabbreviated form,
such as
"IEXT(I(owl:AllDifferent))",
is used instead.

Modified Tuple Notation Style [NOM].
Tuples are written in the form
"( … )"
instead of "< … >",
as in the other OWL 2 documents.

Deprecated Vocabulary Terms [DPR].
The following vocabulary terms have been deprecated as of OWL 2
by the Working Group,
and SHOULD NOT be used
in new ontologies anymore:

[correction] Changed the range of the property "owl:annotatedProperty" from IP to IR in order to avoid undesired semantic side effects from annotations. This was an oversight when the original semantic conditions for annotations of axioms and annotations were removed from the document.

[nonnormative] The semantic conditions and comprehension conditions for the n-ary property restrictions have been changed to only cover property sequences of length greater than 0, since the meaning of an expression with an empty property set is not clear.

[editorial] Explained the optional status of the semantic conditions concerned with the IRI "owl:onProperties", in accordance with the rest of the OWL 2 specification.

This document has been produced by the OWL Working Group (see below), and its contents reflect extensive discussions within the Working Group as a whole.
The editors extend special thanks to
Jie Bao (RPI),
Ivan Herman (W3C/ERCIM),
Peter F. Patel-Schneider (Bell Labs Research, Alcatel-Lucent) and
Zhe Wu (Oracle Corporation)
for their thorough reviews.